I don't know where you live but I feel like most tattoo artists are pretty liberal, they have seen everything.
I had a tattoo done on my chest pre-op, this is what happened. (and I had this done in the south)
He assumed I was cis and said "ok, take your shirt off buddy"
I explained my situation and he said, "oh ok, well I have some sheet napkins that you could use to cover up"
I ended up having him leave for a second and I took my binder and wore it like a "strapless dress" by putting my body through the neckhole.
he was completely respectful and didn't even ask me any of the typical trans questions, he just tattoo'd me.
you may not help this helpful cause you don't wanna let him know you are trans, you could always wear your binder like I did and just tell him you are a little uncomfortable and if you could cover up? Maybe wear a tank and do the same thing with those shoulder straps as with the binder too.