1) A wooden bathtub placed on a radiant floor will dehydrate and crack. No matter if the wood is coated or not.
2) Besides this, coating only one side only is never a good idea as the wood will be unbalanced and start to warp
3) In case of a japanese bathtub, we never coat it as it would lose its aroma. To make the maintenance easier we suggest to use a full penetration water repellent product (Kihada ichiban)
to solve the problem in your case I recommend:
A) to sand the outside of the tub to remove the coating completely
B) To keep the tub always full to the rim while the radiant floor is on. Even when you are not taking a bath
C) If you want, we can sell the water repellent. It prevents stains and the formation of mold.
Please let me know if you have additional questions.