A Mauser action in the white without barrel and with no stock is about what, $275?
I see in the white cz550 as a barreled action in .375HH for $680.
In both cases I'm not sure how you'd get a finished gun for less than $1200 plus dozens of hours of labor assuming you have mill, lathe, reamers, no-go gauges, stock tools, etc.
So the puzzle would be, what in .375 exists that is ugly but shootable as-is so you have all the parts you need, but is frugal. Answer would be a whitworth / interarms perhaps. $650 is as cheap as I've seen them because that's what the distressed used parts are worth.
Not being negative, just trying to figure out what to get you for your price target and I'm not sure?
It's like asking for a Ferrari for $5000. No matter how bad it is, a junk yard will strip it and sell the parts for more than the price target. I don't know how to find you a .375 for cheaper than scrap value but I'll keep thinking.