From my limited experience of using some of the SWS out there , here are my picks for rifles that are next BEST , IF you cannot / will not run a AI .
In 308 class , its a OLD school Parker Hale M85 , uses a M14 mag , cutdown to 10rds
In 338 class , 1st is the PGW Twolf , 2nd the Sako TRG-42 .
The M85 is NOT available new any more , stopped making them around the end of 1989ish , so 2nd hand ONLY for that one .
The other 338s are still made , be aware that they are NOT 10rd double feed like the AX338 .
I will say that you are best served with a AI rifle that comes closest to your needs , and IF you need a cal not made by AI , then re-barrel witha top tube .
In short slightly customise a AI rifle , maybe you will use a different cal/ barrel lenght / barrel profile , maybe a custom bipod of the likes of a revolution on the AX series , and maybe different scope mounting options , and thats IT , move ON .
I am fully aware this is not what MOST want too hear , I too for a LONG time , said AI are too expensive , and custom builds where the way to get the perfect RIFLE , did that for too LONG , and its WRONG and in correct , just like building the perfect 10/22 rifle , by putting customs parts on it , ie replacing almost everything , you are BETTER served by saving up , buying a Volquartsen , thats when you really look at it , again , not what most want to hear .
My experience has been expensive and long , and it boils down too , just saving UP , and buying a AI rifle , and forgetting about everything else .
ME , my AX338 , its going to be modied slightly , with a heavier profiled barrel , maybe fluted Bartlein in 338LM , maybe std or an inche or 2 longer than the std 27inche , looking at trying to use a Revolution type bipod on it as well , and thats about IT .
Later Chris