While it may have some identifying marks under the grips, a serial # wasn't required back then. They weren't made for all that long and not in great quantities, either.

I've only seen one of those in my lifetime, and I seem to recall it having ivory-colored grips. That was back in the early 80s in Austin, TX. There was a small, older store called Chuck's Gun Shop on the I-35 feeder road near the UT campus. The wooden floor creaked when you walked on it, and the place smelled like Hoppe's #9. I am sure I left a portion of my tuition scholarship $$ there (and at a larger, more modern place called McBride's) back then. But to a kid who grew up with virtually no reloading supplies in his home town, those two places were intriguing and kept me broke haha.

That's a nice pistol you have. I hope you can find the parts you need. If nothing else, a good machinist should be able to replicate a firing pin for you.