Hey Jake,
The only mixer that I know of that records is the Sound Devices 552. 5 inputs 2 track recorder built in.
Certainly, I would use a mixer and send a feed to the camera as well as a recorder. Zoom H4N’s are popular inexpensive recorders. There are a number of mixers you could choose. I use Sound Devices 302 and 442 mixers.
SQN,PSC, Wendt, Shure all make mixers; it depends on budget, of course. If I were you I’d rent.
Cheers
Jon