If it's happening with 3 different scopes, then I would say it's more likely to be a bedding/barrel issue. Have you tried letting the barrel get dead cold between shots?
I just realized it has iron sights too.
Pull off the scope bases and go shoot it with the irons. That’ll tell you right away if your issue is the bedding or something in the sight setup.
Are the action screw holes blind? Could the action screws be bottoming out before tightening the action to stock?
Probably not real helpful, but I notice that barrel has a "boss" for the rear sight mount, like the pre-64 Winchesters. Does it have a stock screw up into that boss (again, like a pre-64 Winchester)? I have a number of those BSA conversions here, but all are or were 30-06, (none in .270), and the rear sights were installed with a simple dovetail into the barrel. These were former Model of 1917 Enfields, converted by BSA. As a 270, it should have serial number starting with "HA###x". Since there were no milsurp 270 barrels, I presume that BSA had to get them made somewhere.
It appears to have Weaver bases. From what I've read, the receiver was supposed to be ground for Parker Hale #3 and #4 bases (similar to the Win Model 70 contours). The rear base hole spacing from BSA was .605". So far as I can find, only a #36 in the Weaver line-up has that hole spacing. If that rear base is #36, it has a larger radius than the rear bridge was ground for, and apparently can "rock" under recoil, even though the screws are tight. Might be worth investigating.
My 300 ruger was like that when i got it. Found it would shoot one, the barrel whip to the opposite side, and back to the first side after the 2nd shot. Ended up bedding the foreend and now does 3/4" group.
I dont think its an upwards pressure issue. I think you need to bed the 2 sides of the foreend instead of the bottom.
If it’s a barrel heating/metallurgy/unequal heating issue, then the bullet holes should walk along a line, then settle down into one spot after the first 2 or 3 shots, so I don’t think it’s that.
Op, could you post more pictures of the rifle and sight set up? Not that we could diagnose a bedding issue through a photo, but sometimes staring at something helps with the chin scratching. ;-)
... Hope the fix works for ye Ted.
JJ




























