M.G.
Your option is to work out a deal with your mortgage lender to allow you to forgo putting tax money in escrow. In that scenario, when the tax bill falls due, it is up to you to pay it. This obviously requires discipline on your part. Under what circumstances a mortgage lender will allow you to forgo escrow is something I do not know. Back in the day it had something to do with your financial "worthiness" and how much equity you had in the house. A call to your lender will sort that out for you.
I too am surprised your setup includes the bank giving you the check made payable to the Treasurer. Most mortgagors would require that check to go directly from the bank to the Treasurer.
Good luck.