I've flown with a checked Brommie several times in Europe. Remove hinge bolts/clamps and slide into the standard Brompton slipover cover. Sometimes wrapped some cardboard inside the cover but more often not. Check in as a mobility aid (never use the bicycle word, the check-in drones start following their script) and you are good to go. My earlier Brompton would often have the rear rack bent sideways a little, so push it back into alignment and ride away. Never any other hassles.

A hardcase limits your mobility too much and kills your weight allowance. A Brompton is a tool and I can't get too worked up over scratches, YMMV.