I have been a qualified flooring installer for 12 years and fitting for 16 years. Having moved to a new area recently I’m building my business again in my local area.
I specialise in all flooring after being fortunate to of had a long time with a variant of flooring jobs at high end properties with high end products.
I would suggest finding out what is underneath the felt. if it’s concrete great tile straight onto this maybe put a liquid dpm first for peace of mind but providing tiling is fitted and sealed correctly no problems.
if wood is underneath a water proof board ideally quite thick and strong like osb board could be fixed the the floor and tiled over this again making sure it’s fitted and sealed correctly with the right adhesives and grout for outside use.
My suggestion would be to buy i deeper door bar unless it’s due to the underlay causing the carpet to be higher then the bar causing it to not easily tuck under. to solve this you would need to remove the door bar put something underneath to raise the bar giving better access for carpet. the only other reason i can think of is the carpet has been cut short.
Always where possible tile first as this will give a cleaner look and more importantly give better sealant for longevity of your bathroom furnitures.
In any wet room sealing is crucial so providing all areas are sealed you won’t have issues water causing problems.
if it’s piece of mind that’s more important you need a dpm, screed and water resistant glue for the flooring as well as making all other areas sealed correctly.
Replacing the whole area would be best, i agree with a dehumidifier also but even after that i would suggest getting damp tested in the walls and if all is ok make sure the walls are flat for best results fitt cement boards where the shower will be also the right adhesives and grout are used and then that everywhere is correctly sealed after.