I haven't jumped into the hotswap game yet. With everything going on with COVID, even though I already ordered the parts to assemble my first LEGO board, it seems like that may still be far off. It appears that that plate allows for disassembly of the switch without removing the switch though? That seems like a curious design choice to me for a board that's hot swappable anyway.
I have done a lot of switch desoldering and swapping, in conventional plates, and I have never had this happen, so I imagine that design quirk is the cause. Could you try getting a guitar pick (or spudger if you don't have these superior tools) wedged between the lip of the switch housing and the plate, or use some precision tweezers to try to get a grip on something to try to pop at least one side of the switch out, then the other?