Ditto SusieQ:
Your docs DO need to be communicating with each other. My husband was in a similar situation last year, and it took a little coordinating, but it's do-able (stent placement and liver biopsy, that is).
My husband had a non-medicinal stent put in last year on Feb 1st and started on oral Plavix. He was scheduled for a trans-jugular liver biopsy (as copyman said, lower bleeding risk than the traditional biopsy) the first week of March and was instructed to stop the blood thinners (in his case both Plavix and aspirin) and keep to minimal activity for 10 days prior to the liver biopsy. Biopsy went smoothly.
As for starting treatment with an existing heart condition/being on blood thinners -- that's a question just recently posed to my husband's cardiologist and still waiting to hear back, so I'd be interested to hear what your docs have to say.
it is very important to find out what if any liver damage you have. your complications makes this even more important then someone else. since you can not have a conventional biopsy then perhaps a trans jugular biopsy where there is little risk for bleeding. or at least a non-invasive test like a fibroscan or fibrosure. if little liver damage then you could wait until better drugs come out that hopefully shorten therapy. these harsh tx drugs can be very bad for the heart so proceed with caution. best of luck