I feel like you’re probably making too big a deal out of moveset exclusivity. Pretty much every Pokémon and Fire Emblem character shares several moves with others in their games. Byleth, for example, borrowed the other House leaders’ weapons for their moveset. They might have had enough to work with just using the Sword of the Creator, but they didn’t choose to go that route. Characters like Captain Falcon never had any moveset at all to work with since he was just a driver. To me, as long as the character feels faithful, it doesn’t matter whether they were actually completely unique in their source game. There’s plenty you can pull from that can separate him from the rest of the roster.
Actually, I'm not. I'm following the criteria set by Sakurai.
He didn't outright choose the Ultimate DLC characters. For that matter, every Pokemon after Brawl are a special case and had a slot set aside. That particular Pokemon was something of a special option that he still had to have "dance in his head". Hell, even Brawl's were pretty special. Lucario had debuted at a good time for the franchise, and had a clear fighting style already made, along with a very unique Aura gimmick. Pokemon Trainer consisted of 3 unique Pokemon rolled into one, creating a unique dynamic to begin with, which isn't just battle switching alone, but the Stamina ability(while annoying) was created to encourage you to switch out as a battle mechanic, after is was a failure to actually matter for Zelda/Sheik. Of course, it didn't work as well as intended, but it's very clear how it's made to work. Besides that, the switching out is a pre-set cycle, a mechanic from the basic rpg games, which also makes it stand out. And that's before you get into the Pokemon themselves, whereas Squirtle beat out Blastoise just because it was small enough to create better balance. Which admittedly is just more luck it worked out that way, but it was a pretty good call for a more unique character anyway. That said, the three-in-one gimmick was pretty much the core thing that helped(though the timing did too, as Gen 1 had gotten remakes recently, and thus, there was a lot of focus on the Kanto Starters again).
...It actually applies to Fire Emblem. First, this was after Brawl, where Ike was the only new one added. Roy was a clone first and foremost. Lucina got in cause an easy clone/echo, the core exception to this. Robin was one of the three main characters who was a specific magically tome user, and somewhat of an attempt at breakable weapons, and got in because he couldn't envision Chrom as unique, due to being too similar to both Ike and Marth combined(which also got him in as an Echo a game later, since Roy is practically a cross between Ike and Marth anyway, being somewhat of a middle ground between them, moreso after the moveset update). Chrom, again, is an Echo, so couldn't count for using that criteria. Byleth is the only protagonist among the total of all 4 main characters of Three Houses that isn't tied to a strict house, but also could plausibly pull off said Weapons Triangle(well, moreso as a reference) because of being said middle character among them all, and even then, Nintendo chose him/her, so... it's not exactly like something couldn't be made out of the game. FE is an odd one since it's had weird development for characters chosen, so it stands out as being a non-simple situation. While the gimmicks aren't perfect, they also come with the straight-out caveat that there's very often a bigger situation to why one got in. And it wasn't "cause it's Fire Emblem", same reason with Pokemon. They didn't simply get in cause it's a popular franchise. Set aside a slot(twice), sure. But it's easy advertising for upcoming games and since Sakurai is dealing with The Pokemon Company(who isn't simply one company to work with), his options are often a bit limited as is. Generally not the same for any other franchise.
While there's no denying exceptions happen(generally cause of clones or echoes), they're not actually that often. And surprisingly, Fire Emblem actually follows that overall criteria. I mean, having a unique move does not mean you can create a proper moveset simply from it. And I honestly don't see how the flying carpet makes for a very clear moveset option. I agree the power-ups absolutely can, and doing more from his SMB2USA self is nice too. But it's not based upon a specific ability created for him alone at that point. And as he's in the game based upon a legitimate role he has and does(though not the only one that does), they already have an idea of what his gimmick seems to be(being basically a shield for others). It also doesn't help that there's a clear case that the team doesn't view him to stand out much at all from the other Toads(if him transforming into Captain Toad apparently is the case. I can't find any data on its Japanese name, with only a fan wiki able to say "it's the case". That's not a citing, since it's not like I'm reading the Japanese name of the Spirit, heh), and even if they did, Toad admittedly is pretty similar to the race with at most going out on adventures. But that goes back to the other point of what makes him stand out from the other regular playable characters in the tons of games. And how SMB2USA would be giving him an ability that obviously only he could use(it's hard to say the Flying Carpet should count when it was not exactly based around him at all. His other abilities, being the charge jump and faster item grabbing however might be able to make for something more coherent).
Though keep in mind that with a lot of these others, their cases were a bit more unique, and also weren't in a position where they were only really shown to be there to compliment another character. Chrom was already considered for the playable Awakening rep first, so his later role as a single move(one that isn't as active as Toad is, mind you) was more of a secondary option. There's no telling if Toad was ever considered, but also, Sakurai basically decided to make said Toad a much more active part of Peach's moveset, far more beyond how Chrom is(which many have wanted Chrom to be replaced with another character. I can't remember who was said at the time, mostly cause it was years ago), so he seems to have already made a much more clear decision on the concept now. On the other hand, maybe this is the time he has more ideas for Toad and that was just the beginning. Who could say. Basically, with what we know, what'll make Toad stand out for Sakurai is not a simple move that was used by others, but something that others don't do as of now. He was voted high in the Melee poll, so it's very clear he had to have at least been given a look at. Hell, maybe that's part of why he was added as basically the first time a character summons another one(as minor as the move is), it shows how important he is to Peach anyway. That's not a bad thing.
But anyway, I don't think he's in a great position till they start trying to give him more unique traits in the games. Was there any other skills from the various spin-offs that was specific to him? I remember him and Toadette getting a Mushroom option for the special move in Mario Kart Double Dash(and Toadette for the most part would likely to be an Echo or even a possible alt, of him). so maybe something only they do could be a great start for a very coherent moveset. And unlike some other cases, it's not a matter of a "generic power-up many use", but an actual case where they designated it's "theirs" as a gimmick. I'm not sure how it could work, but that combined with various power-ups, his SMB2USA abilities as well, could make for a very coherent moveset. It does help it's something a bit more specific to him too(or designated that way at least). So something like speed bursts as an attack while combined with other power-ups could make it really stand out. Fans have made a very stand-out moveset for Captain Toad, as well as ideas for having his lack of jumping power(or lack of entirely) as a plausible gimmick to really bring out the creative juices. That said, I'm not against it, I just don't have a huge amount of faith in him being likely, that's all. Coupled with the history of him being a move(and why it makes sense in a way). But yeah. Also to be fair, when I do ask for an ability that's more or less actually exclusive to him, it's hard to find one. Now imagine how any developer would be when seeing a single move(which technically, he doesn't even have) and what to do with said single move to create a full-out gimmick for a coherent moveset. Consider also that while the Mushroom thing I mentioned is a designated special ability for him, it also isn't necessarily an obvious gimmick to use(as something like quick speed bursts is surprisingly hard to implement in games as a gimmick. It's generally easiest to do in media where it's not about gameplay, but some kind of cool fighting style. While possible, you have to balance out how he fights when not at full speed, and how long said bursts last, etc. Basically, this is a great idea, but it also needs to take time to balance out. You can imagine that for any such move, some being harder than others. Assuming no programming problems(Dixie's tag teaming with Diddy was scrapped due to being unable to figure out how to make it work) of course.