Hang the beadboard so that it comes down and meets your bullnose trim.

Then rather than taking off the bullnose trim, purchase some MDF trim and cover the tile completely and have it come just slightly higher than the bullnose so it covers it and meets the beadboard.

Here is a link to what I'm suggesting:

MDF trim

Caulk the trim where it meets the beadboard and use an adhesive to attach the MDF trim to your bullnose tile.

The MDF trim can be painted to match other trim in the room or it could be painted dark grey or black to match the floor if you want it to blend in more.

I found a picture that isn't great but shows what I'm trying to describe. Imagine your bullnose in behind the square MDF. Caulking where the trim and the beadboard meet.

beadboard with square baseboard - Google Search · More Info

M V... thank you so much!