I've been dealing with a similar situation.
I have an older 700 BDL Varmint. It has a 1-10" 24" varmint barrel. In the past I've used it to shoot 100 gr BTSPs on mule deer, and it has worked very well for that out to 300 yards. It shoots lightweight 55 gr varmint bullets sub MoA out to 800 yards on no wind days. My shooting is mostly at about 5000' and above.
So, recently, I've been wanting to stretch the range a bit, shooting steel targets out to 1000 and 1200 yards. I tried Nosler 105 CCs, and Hornady 105 AMax and BTHPs. Out to 800 yards or so, all shot well. When I pushed out to 1000 yards, and 1200 yards, I would get a few hits, but lots of wild, unexplained misses. At 1200, I finally found out why.
I hang my steel targets from 2x4 wood sawhorses. At 1200 yards, I found several keyholed bullet holes in the 2x4 supports. Evidently, the bullets were destabilizing and were tumbling. They shoot ok inside 800 yards, but beyond that, no joy.
I bought some 87 gr VMax, but haven't found a load the rifle likes yet. And at distance, they are pushed around by the wind much more than the 105s were.
I am currently working on loads for 95 gr TMKs, and they look promising. I'll get a chance to try them at distance in a couple of weeks; I'll update then.
So, when you ask if you can shoot 105s in your 1-9 1/8 barrel, it depends on your altitude, the velocity, and apparently, the distance.
But from my experience, save yourself some grief, and keep to 95 gr and lighter, or use them at shorter distances.