I expect he would show up to Westminster, take the pledge and not wear the costume. Maybe they'd make a big thing out of refusing to let him enter with the Bin on and so forth but maybe it's not worth the bother.
If they did it would be staged, like the modern (and perhaps ancient) slamming of the doors before Black Rod during a State Opening. It's just symbolic, they're going to allow Black Rod in, he did not have an inherent right, as a symbol of the King, to enter, he is knocking to get permission.
I think there is some rule about not wearing hats that block the view of others or something. From the era of top hats and all that which aren'tthat far removed from the bin.
If you want to see how exactly a Westminster Parliament would stop duly elected MPs who did not follow the rules, look at the 2022 controversy about the Parti Québécois MPs who refused to pledge allegiance to the monarch. They were physically stopped by the sergeant-at-arms.