If by “well maintained grass” you’re talking about a flat, maintained sports field or something of the sort, likely yes. If you mean grass that’s been cut but the ground is otherwise natural with bumps and holes, then you’ll probably have problems.
It does depend on the chair, but most are not built for off-roading. Also will depend on if you’re the pusher or if you’re talking about propelling yourself (as the wheelchair occupant).
Don’t know about ‘battle royal’, but are you talking about Battlebit?
https://store.steampowered.com/app/671860/BattleBit_Remastered/
Or in the battle royale genre, there’s TAB:
https://store.steampowered.com/app/823130/Totally_Accurate_Battlegrounds/