First off, there are a LOT of Nintendo Studios - including second party ones, so I'll just list off the ones that make the most famous or currently well known Nintendo games. And feel free to correct me if I'm wrong.
EAD Tokyo: Made the Super Mario Galaxy series. So Shigeru Miyamoto would provide a lot of assistance if this studio made it. He does help out in other games (Paper Mario: Sticker Star....)
That EAD team that made Star Fox: Could be EAD Tokyo in fact. This might the most obvious, but they haven't made Star Fox in a while! What gives? I want a dog fighting games with Foxes and animals again!
Intelligent Systems: I think they are most famous for Fire Emblem, Golden Sun, Advance Wars, and Paper Mario. Er, they're good at RPGS and SRPGS so maybe not them....
Monolith: Makes Xenoblade and Wii U's X . Might not be the best because they only make RPGs every since they were acquired by Nintendo, but the Mechs with Guns in X seemed promising. And it seemed like the game can be online! So who knows? 
Retro Studios: So far Metroid Prime and Donkey Series, and also known to help Mario Kart 7 (retro tracks i believe - which is quite a punny if true!) They have experience of shooters in Metroid and have a multiplayer for Echoes and I guess Trilogy (although that's a port, but allows moving and AIMING in multiplayer). They are also a western developer. But they never made a game online before and are quite small compared to the average Nintendo studios which are also quite small.
GameFreak: Famous for Pokemon. I know! They're usually known for Pokemon which are always 2D with the exception of X and Y, GameCube games, and the Stadium games co developed Genius Sorority. But have you played X and Y and used the PSS systems? I may be wrong, but I believe GameFreak is the most advanced in social networking in games for Nintendo.
Mario Kart Team (forgot their actual name): They always make Mario Kart and probably something else, but usually Mario Kart! They did a good job with Mario Kart 8's graphics, can make massive levels simply because they can design huge track levels, and they are also skilled in using online features like Mario Kart Wii and 7.
Sora Company aka Smash Bro. developers: They're known for the Smash Bros series and Kid Icarus. Kid Icarus Uprising has some sort of online multiplayer and I believe it plays sort of like a FPS because Pit can move and shoot? They may be one of the biggest Nintendo studios even though they are second party, but only when make Smash Bros. But the idea of Smash Bros. character in a FPS game sounds interesting right? Mario with his fire flower, boomerang, and hammers, Pit with his arrows, and the space dudes with their guns.... Maybe this should be a new Nintendo series itself?
Zelda team: Probably one of the biggest Nintendo studios in size and recognition for one of the most prestigious gaming series since the 1980's(or 90's?) The faults: They usually make Zelda games. The Pros: Large number of people who are given the luxury of time to do whatever they want (Skyward Sword, Twilight Princess, and Ocarina of Time averaged 3-5 years of development I believe) and the wisdom of Eji, and Shiggy.
There are probably more like Monster games, Next Level, Brownie Brown, Hal Laboratories, and etc, but I have class soon. Feel free to mention them in your post if you think they should make a Nintendo FPS game though!
