Heck! Now that I've posted this I seem to be doing everything I should have done first. Just paint me DUH!
Anywayyy...

I wrote Ithaca Guns. Even though under new ownership, there is a continuity with the ould company. They replied (in less than an hour I might add). Here's what they sent.
Ithaca Gun Company Customer Service to me

Dear Jim,
During WWII, Ithaca was focused on producing the Model 1911 A1 Pistol for the U. S. Military. Ithaca did produce Model 37’s between 1944 and 1945, but they were not kept track of as far as the serial numbers. Your Model 37 would have been manufactured between 1944 and 1945.
Thank You for contacting Ithaca Gun Company
So there we have it.