Another IFR "oops"

Another one of those things I guess I should have been ready for sneaked up
on me this morning.

We were going from BFM to EET (Shelby County Airport, Alabaster, Alabama)
and the ceiling was right at minimums for the RNAV 33 approach, and below
minimums for the VOR-A. Naturally, I wanted the RNAV 33.

There's no ATIS at EET, so no particular approach was being advertised. When
Montgomery approach handed us off to Birmingham, BHM approach told me to
expect the VOR-A. I asked him "...any chance we can do the RNAV 33?"

"Nope. It's not in my airspace. You should have asked Montgomery for that
one. Standy by and I'll see if they'll take you back."

We got handed back to MGM, but by the time MGM was ready for us, we were
right on top of the IAF (IXUSE) and 1,900' high. Not a serious problem,
really, because by then I had slowed to 90 kts, but it made the first part
of the approach more rushed than it needed to be. Nevertheless, we made it
in just under the cloud deck

The interesting thing is BHM's comment that I "...should have asked
Montgomery for that one." How was I to know that? BHM is named on plates as
the approach facility for EET. There's no ATIS at EET. What should I have
done to find out that BHM couldn't give me the RNAV 33 approach?
--
Dan
C172RG at BFM