Xenia, if pre-anesthesia blood work was done prior to the dental, then basic liver values would have been included. If they were elevated, I'd be surprised if your vet went forward with the dental without first investigating the cause of the liver issues.
A good next step would be a "sit down" with your vet to ask what he suspects as the cause of the elevated BATS. Elevated bile acids can be caused by a shunt or MVD (many little shunts within the liver) but there are other causes such as hepatitis, gall bladder disease or sludge, or problems with the pancreas. An ultrasound (with high resolution equipment and read by a board certified radiologist rather than your regular vet) is a good non invasive diagnostic tool and can provide information about the liver, kidneys, pancreas and gall bladder. Ultrasound guided needle biopsies of the liver are not terribly invasive but often come back with inconclusive results, but an open liver biopsy is major surgery, so I would be asking tough questions before agreeing and would want that type of surgery done in a specialty hospital and by a specialist.
If your dog appears well and is eating and otherwise acting normally, you might consider talking with your vet about changing your dog over to a low protein liver friendly diet and treating with supplements for 30 days or so and then repeating the bile acid test. Many types of liver disease are managed well with diet changes, supplements and medications, and many dogs live for years with well managed liver disease.
Both milk thistle and samE are good supplements for the liver. I used Denamarin for my dog. Denamarin is one pill that is a combination of milk thistle and samE. I ordered it from Amazon (no prescription required) at a good price and the convenience was worth it for me.
A helpful tip is to always ask for copies of all lab work and test results, and then to keep them in a file. My vet just emailed the copies directly to me. If your pup needs to see a vet on a weekend or at night when your vet is not available, it is useful to have those copies handy to take with you.
At this point, your vet is your best source of information. These are just some suggestions to bring up when you talk with him. I always found that the more informed I was when I talked to my vet, the better the conversations went.
Good luck with your pup, and please keep us posted.
Diana