You actually really ARE allowed to wear a talis godoil in the bathroom. (We don’t because its not kovod but halachikly there is no problem).
That is the reason that one doesn’t make a new brocho on a talis godoil if you take it off in the middle of davening because it is not cosidered a hefsek because its mutar to wear it in the bathroom.
Whereas tefilin which is a dovor shebikdusha is ossur to wear in the bathroom and therefore if one tales them off in the middle of davening to go to the bathroom it is a hefsek and you will have to make a new brocho.
Therefore it comes out definetly halachikly there is no prob wearing a hat in the bathroom. And I don’t think there is a problem of kovod either.