Primed and painted MDF baseboard will be fine in a bathroom or anywhere else. You can't go crazy with your mop though.

Can you find a composite plastic waterproof baseboard that matches? If not, I think they make a wide flat casing in that material that might look nice. You can paint these to match your other baseboards. I found 3” baseboards in composite plastic that were identical in profile to the generic wood and mdf baseboards I had in other rooms. If you use the flat casing, you should be able to rip it to the 5” size you need.

I think you should replace the mdf baseboards in the bathroom. I had wood baseboards in my bathroom before I remodeled. They were the motivation for the remodel (water snuck out the shower door and that corner got wet). I don’t even use mdf for baseboards in the basement.

I would think the kitchen would be OK.

Water will find it's way under and behind the mdf. Change it out, put composite or wood.