I actually see this the other way around - I see them painting the stages, but not the characters.
It's more important to the Inklings' mechanics, first of all. The stage would have to be painted in order for them to be able to swim or use stealth (and it's why I'm stuck on "break the media" actually meaning "break the meta," if there's any truth at all behind those pesky tweets - they'll completely redefine the zoning game, at least while they're on stage). They'd need to take some liberties in implementing them, like severely reducing the time an area remains painted to a few seconds, but it could be cool. Painting the stages shouldn't be terribly difficult either, so long as they restricted themselves to static surfaces.
Painting characters, however, has no benefit in a Smash context. The point is to rack up damage, not completely paint the surface of an enemy. More importantly, applying paint effects to every character model could be incredibly challenging - it's easier in Splatoon because the models are relatively simpler and more similar, but imagine trying to properly apply a splat to Ganondorf's armor. Painting each character would be purely an aesthetic choice, and one that seems like it'd come at a high cost for little payout. Maybe it is made simple through some cheeky algorithm, but you've asked the right question - is it feasible on a 3DS?
Yeah, that's why I figured paint attacks on enemies might just have a splash effect and nothing else. I understand the whole gimmick is painting the landscape but at the same time the only weapons Inklings have for attacking their opponents are also paint based. I just imagined the full body texture of the metal box for instance being easier than spot painting an enemy, if they were to try and make paint stick on an opponent for any reason.
I'm not sure how paint textures on stages would work either. Again, spot painting stages seems like it could get messy in more ways than one if it wasn't limited. And by limited I mean I don't think it could last too long or be too big, not just from the standpoint of gameplay balance within Smash but also what the 3DS engine could manage. Basically, if it was very limited it could work. Rather than existing as a "retexturing effect" it could just exist as an attack effect similar to any other attack effect.
I also imagine stage painting would only be a narrow stream or spot existing only on the "fighting plane". I don't see the effect having any variation mostly due to the 3DS, as in the paint effect of a move would appear the same every time. With the exception of say a special that leaves a spot for several seconds as a place an Inkling could have more "power" or whatever, I think most ink effects would only last as long as the attack animation. For instance, a move where the point is to have a spot on stage for at set amount of seconds could be similar to the Villager's sprout and could have the effect of increasing the power of the Inkling while in that area, or cause harm to enemy while they're in it, or just make them trip or something. The limitation would be that you couldn't do this move again until it wore off from the first time. I also see an issue with multiple Inkling's on the stage as well, but if just one move left ink for any length of time it could be not too messy.
Other ink based moves may not paint anything. Say the neutral special is just shooting an ink gun. I could see that move simply being a projectile that has a splash effect when it impacts an enemy for damage. After the splash there would be no ink left anywhere. Likewise, a squid jump special could have an animation where the Inkling creates a stream on the ground or in the air and passes through it. Think Pac-Man's side B but appearing more like Shadow Sneak or something. After the move was made, the ink would be gone.
TL;DR - I see the Inkling's being doable but they'd be much much less messier in Smash than in Splatoon. I realistically only see ink sticking as part of one or maybe two moves. All other moves would just have splash effects and wouldn't leave ink behind. The theory being that this kind of limitation, while certainly not accurate to Splatoon gameplay, would be a lot less taxing on Smash's game engine by making things a lot less messy.