just to point out that socket outlets are not allowed within 3m of zone 1, which is the edge of bath or shower...,
Whoops - my mind was obviously thinking (incorrectly) of 3m from zone 2 - hence my 3.6m figure. However, my point remains, even with 3m, that not many bahrooms in standard houses (and, one assumes, certainly a bathroom with no wall space to fit a towel rail!) will be big enough for sockets to be permissible anywhere.
... if closer then a fused spur adequate for the location could be used.
Indeed - that was obviously the point I was making. The OP's plan to use a flex outlet plate with the FCU outside of the bathroom is probably even better.