Before I start, feel free to tell me if I'm doing this wrong - I've not really done a moveset submission for a character before now on the Infinite thread, and I've noted that a lot of other people here have as of yet only been submitting individual moves - so if this is a taboo, I deeply apologise in advance. However, here's my thought process.
Stats 'n Stuff
Firstly, I think it'd be pertinent to observe Saki's stats - it'll be a good way to get a feel for everything surrounding them. With that in mind, I'll steal this basic layout from Delthezin:
Stats:
Weight: 78, equivalent to that of Sheik and 84th heaviest overall - a clear lightweight.
Walk Speed: 1.42, equivalent to that of Mythra and 8th fastest overall. Good for positioning.
Initial Dash: 2.45, again equivalent to that of Mythra, and tied for fastest overall, making it good for burst.
Run Speed: 2.31, equivalent to that of Zero Suit Samus and 7th fastest overall.
Air Speed: 1.11, equivalent to that of Fox and 33rd fastest overall.
Fall Speed: 1.75, equivalent to that of Sheik and Diddy Kong, and 24th/25th fastest overall.
Ratings:
Size: 6/10.
Frame Data: 7/10.
Damage: 4/10.
Combos: 6/10.
K.O. Power: 5.5/10.
Recovery: 3.5/10.
Furthermore, Saki has the ability to wall-jump - but not to crawl.
As you can see, this paints a picture that meets the suggestion originally made - a fast-paced lightweight with surprisingly decent jump height, coupled with a mix of quick attacks up-close and pressure tools from afar - though these good points are offset by lower weight and air speed. I also think it'd be a nice alternate take on the genre of the rail shooter compared to Fox, with Saki's moveset moreso actively rewarding a mix of zoning and pressure whilst being mindful of your surroundings. Unlike more full-on zoners, however - like Samus or Mega Man - Saki's use of projectiles shouldn't in itself be a K.O. tool, but should be a method of putting pressure on an opponent and allowing for his close-up attacks to really hit hard. If anything, I'd say he's a mix-up character overall.
Prominent Attacks
Admittedly, I won't cover all of his attacks in detail - for the most part, they'd rely on either basic kicks or slashes with the Dolphin Gun. However, a few attacks felt like they'd be important to delve into.
Side Smash: Overhead Dolphin Slash.
A powerful slash with the Dolphin Gun's blade, which moves from close to Saki's head down to his feet - before being able to be cancelled into a second, slightly less powerful attack that has him slash forward in front of him with significantly less vertical coverage. Notably, there's a sweet spot that allows either of these attacks to reflect projectiles that hit it, just like in the original game (and somewhat comparable to Ness's Side Smash.) It's a little counterintuitive, because the sweet spot is on the gun itself rather than the blade - but with a little bit of getting used to, you'll get the hang of it.
Overhead Dolphin Slash is arguably one of Saki's most powerful moves - working decently well as a kill move, but with the necessary charge making it a little difficult to use in his typical mix-up gameplay... even if the second hitbox lets you potentially draw unsuspecting opponents close for a surprise attack. I'd also like to note that I considered giving him a reflector down special, but it didn't feel like it was keeping to the spirit of the original game - where reflecting a projectile required more skill than just reading it - so I introduced it here instead!
Neutral Special: Manual Shot.
Tapping the special button allows Saki to fire a shot straight in front of him, which isn't aimed in any way. It functions similarly to Fox's neutral special in that it's a low-powered projectile that deals no knockback - though Saki's has some differences. Firstly, Saki's Manual Shot doesn't diminish in attack power with distance - having a flat 2.2% damage rate regardless of where you fire it from. It also has a slightly faster rate of fire than Fox's if you repeatedly tap the button. This form of the neutral special is pretty unique in that it's low in lag, allowing it to be used as a decent walling tool for pressure. Not only this - but if you hold down whilst repeatedly tapping neutral special in the air, you'll be able to shoot downward at a 45 degree angle. Simply stop tapping the special button, and you'll be able to use your down special should you wish to.
I should also note that the move is chargeable, though this charge is really quick - only around 30 frames. Holding down the special button until a visual flash from the muzzle acts as an indicator allows you to fire a version of the move with very slightly increased power (around 4.2%) as well as making an opponent flinch, though this comes at the cost of being a little laggier overall, and not being as easy to quickly fire off (even though it does fire automatically when at full charge.) Again, if airborne and holding down after starting the charge, you can fire downward at a 45 degree angle. This specific function of the attack is in reference to the ability to charge your projectile attacks in Star Successor, though keeping Saki feeling quick and adaptable overall and not getting in the way.
Side Special: Lock-On Shot.
Inputting side special has Saki ready the Dolphin Gun in front of him. When in this state, the closest opponent within a certain range in front of him (around half of Battlefield) gets a reticule over them, and Saki fires a single projectile at them. The Lock-On Shot is a little more powerful than his uncharged Manual Shot due to its more deliberate nature, dealing 3.2% damage and making an opponent flinch. However, the animation itself is a little slower, and it's not as spammable due to needing to lock onto an opponent each time it's used. Not only this, but it has a very slight recoil when used whilst airborne - as well as looking very slightly different. I should note that the range itself has similarities to Palutena's Autoreticule in that it's more akin to a "cone" outward, preventing him from simply firing at opponents over him - though, the cone is a little wider and allows for wider coverage as a result.
Up Special: Dolphin Burst.
Based loosely on Isa Jo's hover-backpack in Star Successor, Saki instead has a small burst of energy from the Dolphin Gun which allows him to move upward at a diagonal angle. During this angle, Saki draws his Dolphin Gun and uses the blade function to multi-hit opponents that it contacts, dragging them along before a final (optional) slash at the end can knock them away - working a little like Shulk's Air Slash in that you can activate it with another tap of the special button. You can influence your direction a little by holding forward-up for more horizontal but less vertical motion, or backward-up for more vertical but less horizontal motion - though, be careful, as using this move will leave you helpless until you either touch the ground, get hit, or get killed. Compared to some other moves out there, its range isn't the best in the world - but it's functional as Saki's only real recovery option, and can work as a kill move in certain circumstances.
Down Special: Evasive Manouvres.
This move is a little odd at first glance, because it can have one of multiple effects depending on the context of where it's used. Put simply, it works similarly to your traditional counter move - though is designed to better-reflect how this would have worked in Sin and Punishment. When inputted on its own, Saki will apparently spot-dodge, albeit with his Dolphin Gun drawn. Then, right after spot-dodging, he'll quickly fire out a charged Manual Shot with significantly lower starting lag than it'd otherwise have, to the point where it's likely to be safe on most attacks.
You can also hold down-back to dodge-roll backward before cancelling into the shot - or hold down-forward to dodge-roll forward, turning you around if the closest opponent is now behind you, and then letting you fire. In the air, you'll spot-dodge no matter what - sorry, no directional air-dodge here. Note, however - repeatedly using the move will increase its startlag over time, similarly to how actual dodges work (and to prevent it from becoming a method of obliterating the usefulness of neutral special altogether.) The move's unique in that, functionally speaking, it works somewhat like a fusion between your traditional counter and a dodge - allowing you to quickly react to an opponent as opposed to just having an automatic attack right afterward. Just like in the original game, seeing an opening means you need to take the chance to escape punishment and act of your own accord - it's the difference between life and death.
In Closing
I think the moves and stats I've suggested here would work well to keep to the original vision that the submitted had in mind, whilst innovating on other concepts that've floated around the internet for some time. Admittedly though, I kind of like Venus's proposal for a Final Smash, so I see no need to change that. My general focus is on Saki feeling nimble when grounded, but a little out of his element in the air - having access to effective, but not broken, ranged tools that are designed to primarily facilitate him getting in as opposed to just walling an opponent out with an impassable barrier of instant death. Again though - feel free to let me know if I've missed the point here.