If length is same, I can't see the antenna as being overly finicky... For a temp measure I'd repl the coax, then cut the insulation and braid off the the outside end, leaving approx the same length of the center conductor exposed as antenna...
If the housing the receiver is mounted inside is plastic, I'd think the antenna could remain inside... As I stated earlier, the newer garage door openers use a piece of wire hanging out of the circuit board that works fine...
As a last resort you can install a replacement receiver/transmitter that is generally used to update garage door openers with obsolete remotes, I have a Sears branded kit on my mom's opener... They are sold at any of the larger stores...
http://www.ebay.com/itm/Sears-Receiver- ... 1c21f9213a Tom