I doubt the Coast Guard has an "approved doctor list", at least for the basic physical exam. And especially doubt they would (or could) force you to use their choice of doc. Most of us have schooled our own docs in filling out the multipage form. I schedule two appointments--first for the physical exam, second for us to lay out the paperwork on the exam "table" and fill it all out. You gotta pay the piper here.
And on the renewal tests, isn't the rule if you have enough sea time in past 5 years you don't have to take the open-book tests on seamanship and rules of the road? I, of course wouldn't know since I have a weekday office job and never enough sailing or power vessel gigs to make the quota, so always do the exam booklets.