Really just how it is. Unless you can find a lower set of rings you will have that problem. I have low rings and a 20moa rail on one of my Creedmoors and with a 50mm objective I still clear.
I have a stainless stalker in 270wsm. With some handloading it has proven to be a very accurate rifle. Have had it for about 6 years or so got it when I was 15. Took my first deer with it and it still gets used every season.
I do believe swampworks makes a jlk in .257, if I recall correctly the bc is slighty higher than the berger. How true that is I don't know but never the less it is still only 108gr.
SHV scopes have different glass than the NXS line. I had a shv 4-16x56 for a short time. Great scope for the money, very comparable to my Leupold vx-L's at low light.
I had a 1 in 9 twist in my TC pro hunter. I never tried anything lighter than 120. It shot the 120 amax factory ammo great also liked the 129 sst. It shot the 130 bergers decent the 140 amax it would not stabilize. Just my experience I am building a rig now with a 1 in 8 which seems to be the go to.