I would Title the account:
Joe Moe MD Irrev Tr (However the trust is titled)
Trustees Name, Trtee
As an Irrev Tr it should have a separate TIN (remember an Irrev Tr can exist after the grantors death, but you can not report to the IRS under a deceedants TIN). I would not take copies for the trust papers as I believe the institution can be held liable for any and all contents of the trust papers (even if you only take a copy of the first and last page)
I would instead have the signor sign a Declaration of Trust verifing the title, trustee, sucessor trustees, and TIN.
Opinions do not necessarily reflect those of my employer nor are they legal advice