I believe the 37 barrel was desired over the pre A and A 52 barrels, as was the rest of the rifle. I certainly wouldn't go that direction.I have a 52 of about 1935 vintage. It has a Remington 37 barrel. After some digging I found that some shooters preferred the Remington barrel to the stock 52. Both are target type rimfire. My Remchester shoots well, better than I can. This is my only venture into a 52. I have always wanted one but didn't want to pay $1000 and upward for one. This was a very clean rifle and the cost was $525.
As a plus it came with a Vaver / Vartek aperture sight. I understand this is a premium addition. I have added a Lyman Super Target 20X on external mounts.
I just shoot it for fun and use it to test various brands of rimfire.
This might not answer your question, but once I started typing I got carried away.
What you probably have is someone rebarreled a model 37 and the smith then used the 37 barrel to resurrect your 52.