Something like this?

Gamecube had pretty low power draw, so if it was limited to 10W, technically it could've been viable option for Switch like portable.

That said, batteries of that period would be huge problem - you could either go with Li-Ion, which were cylindrical, so you would probably need to put them inside handgrips, like in Ally ROG or MSI Claw (so forget about Joy-Cons), or go with, back then, very new and expensive Li-Po batteries that could be shaped to fit inside device.

Again, all that would cost arm and a leg.

Essentially yes, though NEC was already doing this years prior anyway

Would a Nintendo version of this realistically have seen much success though?

I mean, Gamecube itself already sold poorly, I don't think making it semi-portable would have done much to change that.

I think it could have had some success yes, if they could have gotten it down into that kind of a form factor. 

The GameCube sold poorly because the market was oversaturated for the same kind of product ... home only game console. GameCube Portable would have massively differentiated the system and I think people would have been blown away with being able to play games like Metroid Prime, Smash Bros., Resident Evil 4 on the go, even 3rd party games like Madden NFL, SSX, various Bond games, Time Splitters would've become more valuable in that case IMO. 

The other thing is they could have integrated the Game Boy Player into such a device too so it could have played Game Boy Advance games too. 

What of load of revisionist crap from them - there wasn't anything in the off the shelf market at that point that they've could've gone with - both Qualcomm and Apple had chips in similar performance bracket, but nothing really much above it.

Would such a device really be seen as a viable option for "on the go" gaming though? A device as powerful as the Gamecube would still be pretty bulky back then, and could've ended up in a similar position to NEC and PSOne as a neat little novelty without much mainstream appeal.

Don't think the Tegra X1 was a very efficient chip, it did have high peak performance but it wasn't tweaked well to utilize it, the 20nm process in particular was poor. 

Those were early days for those kinds of chips too, Nvidia had done the K1 which also had overheating issues and X1. 

The more bespoke Tegra T239 which Nvidia made with Nintendo I'm sure is more tuned to specific performance Nintendo wanted, you can already really see this as it can handle PS5-tier games much more easily than the Switch 1 (Tegra X1) could handle PS4 games it looks like. Nvidia is a far, far bigger company these days too so they likely just have flat out better engineers than the era of 2015 or so. 

Nah, they needed the reality check of the Wii U. Both in terms of realizing how much it would cost them to keep two developing branches during the HD era and to actually try and give people what they wanted after the financial and popularity disaster that the Wii U was.

X1 performs just fine, as overclocked Switches out in the wild show.

Only other realistic solution was to go to Qualcomm and get Snapdragon 820 with Adreno 530 (which performs about the same as X1), since there's very little chance Apple would give them their SoC for reasonable price. So, they went with what's available, at best price.

Yeah, probably if they went with some custom design it would perform better, but that would be R&D they had no idea whether or not will make returns - in retrospect everyone's a visionary, but market is fickle and you can't really predict future, so I don't think they were ready for that.

But after success of Switch, I think they had all the necessary info they needed to go to nVidia and ask them for custom chip.