Most of the issues with the belted case are much more myth then actual problems. Once a belted magnum case is fired one and as long as its resized correctly, it will headspace off the shoulder, not the belt, making it exactly the same as a beltless case as far as accuracy and consistancy.
As far as feeding problems, I would like to seen an example that proves the belt makes feeding a problem. Never have I seen this.
There really is not alot of optione between the Win Mag and RUM in a beltless case but there is one, the 300 Dakota. Its a fine chambering. The full equal to the 300 Wby Mag but shorter and beltless. Brass is not cheap but for a bit game rifle its not expensive to the point its a real problem. If you do go with the Dakota, I would recommend you get a good supply of brass on hand.
If your not opposed to a muzzle brake, the 300 RUM is very hard to beat for a long range 30 cal. Its advantage is not in hyper velocity but with the ability to drive long, high BC bullets to very good velocities increasing the ballistic potential of the 30 cal bore.
For your needs sounds like the 300 Dakota may be about your easiest option.