Twenty five years ago, I installed something quite similar (not nearly as nice as your piece) and it has lasted just fine until now. This was installed in our larger bathroom which contains both a separate bathtub and shower. Before installation, I applied approx. three coats of clear Watco danish oil. I allowed the finish to dry about 3-4 weeks then applied 3 to 4 heavily-cut, thin coats of orange shellac. This was partially due to the effect I wished to achieve from orange shellac. I know all will not agree that shellac is suitable for a bathroom cabinet but we never had much in the way of wet and foggy conditions (usually) and the finish has held up quite well. I found the combo of Watco Danish ("Tung") oil and shellac to work quite well overall. It was also dead simple to do prior to installation.
Best I can advise is let it dry a few weeks and then try a top coat such as shellac or some type of varnish (diluted) or perhaps a lacquer. I just found shellac to be the simplest while still giving a nice result for a lot less work.
You can also just apply another 3 to 4 coats of Tung oil. This will take a while to dry but it has to be the simplest way to seal the piece. The more coats of oil, the heavier top coat (and protection) it builds on its own. It takes quite a bit of "Tung" oil to finally create a decent top coat that will handle moisture. Whether you use multi coats of "Tung Oil" or an oil and top coat finish, you can always add a couple coats of Howard's Feed'n wax as a top coat to add extra protection and just plain add a bit of depth to the finish
Hope this helps in some way. You have a beautiful cabinet there, it looks very, very nice in the place you are installing it. Experiment with your finishes. Try mine or try one of the above suggestions. You'll find what works best for you.