We have one 7937 that will only register to SRST, bypassing the CUCM completely, (CUCM v6.1.3). This is the only phone in our enterprise having this problem, as well as the only phone on that network segment not registering with CUCM, so I believe it is something specific to the 7937 model (not routing/ACLs/etc.) It is running the latest firmware as well.
EDIT-The log files off the phone show that it is trying to register with the CUCM's DNS name 'CMBE' not its IP address. It then fails out with a DNS error.