5e RAW, the monk can't use any martial arts while wielding non-monk weapons or a shield, or while wearing armor. RAW, unarmed attacks can be with any part of the body: "a punch, kick, head-butt, or similar forceful blow."
In BG3 the relevant questions are:
1.) Does becoming proficient with weapons make them into "monk weapons"? Signs point to yes.
2.) Does becoming proficient with shields mean that monks can still use martial arts while wielding them? Based on the above (and the loss of multiclassing restrictions), I'd guess yes.
3.) Does BG3 properly implement the 5e RAW monk restrictions? This is unclear. I'd like to say yes, but I could see the Bonus Action Unarmed Strike being usable no matter what (similar to how off-hand attack doesn't require you to make a main-hand attack first).
4.) Will you be able to make an Unarmed Strike while your hands are full? Also unclear. I'd guess no, just because it's not *that* common knowledge that you can kick/headbutt/etc as your unarmed strikes.
In summary, I think that being proficient in weapons/shields will allow you to use all martial arts, but you might need to have a hand free to use unarmed strike.
Yes, knowing the rules would be nice.
