Re: "I have been thinking about your heat concerns..." - agreed. Honestly, that fan grip is massive overkill for most. I've shot in some nasty heat and the standard Canon BG-R20 Battery Grip works perfectly. For the desert, I'd just recommend getting a SmallRig 4567 Cage for Canon EOS R5 Mark II if you want extra protection and heat dissipation. Much more practical than the cooling fan for your budget, tbh... just keep it out of direct sun.

TL;DR: Just get the Canon BG-R20 Battery Grip for EOS R5 Mark II. I am super satisfied with mine; it handles heat fine and fits your budget. No complaints here, honestly.

I have been thinking about your heat concerns since I dealt with some nasty overheating on a shoot last July. I would suggest you be careful before jumping for the fan version though. The Canon CF-R20EP Cooling Fan Grip is the official solution for long bursts, but honestly, it is gonna push you way past that $350 limit. I found that for most stills work, the standard Canon BG-R20 Battery Grip for EOS R5 Mark II works fine if you just keep the LCD screen flipped out to let the back of the body breathe. Also, make sure to only use the genuine Canon LP-E6P Rechargeable Lithium-Ion Battery inside whatever grip you get. I tried some older batteries in a similar setup and the camera limited my high-speed shooting modes almost immediately. If you are just doing photography bursts, the fan grip might be overkill for your budget. Stay safe in that desert heat.