I think it depends on the actual retailer. My local BMW retailer was more than happy to debadge our previous X2, however, my local Nissan dealer outright refused saying “You need to speak to Motability but I don’t think you can. It goes against their policy and I don’t want to get into any trouble”.
Motability cars:
2015 RR Evoque AWD 2.0 TD4 SE Tech - written off
2017 Toyota CH-R 1.2T Dynamic CVT AWD - written off
2019 BMW X2 2.0L Sport
2023 Nissan Qashqai e-Power N-Connecta