I wasn't really asking what rifles you own, but as to what your experience was shooting rifles. 1200-1400 yards can be done without going to the .300 WM or .338 Lapua.
1000 yards with 7mm RM
7mm WSM 1400 yards
2000 yards 7mm WSM
I realize these video's have the 7mm WSM going past 1000 yards, but the 7mm RM and the WSM are similar in capabilities. So if it were me, I wouldn't invest in a new rifle yet. I'd take that 7mm RM Browning of yours and use the money you want to spend on the rifle and buy good optics and reloading gear. Strech the legs on it and then possibly invest in a larger caliber rifle later on. You're looking at a minimum of $2.5K investment in a decent LR rifle and scope before reloading equipment. It's your money and your call but that is what I'd do before going after a rifle in .338 Lapua.