Just to compare. Let's say you had a custom 9.3x62 built on an M98 or maybe Montana action. For your budget you could have a pretty decent rifle, nothing too fancy but a very solid rifle. It would fit you just right, have the correct (for you) barrel length - IMO 21"-22", have a very good set of open sights, a very good trigger, EAW quick detach mounts and a good quality scope.
The recoil on an 9.3x62 with 286gr bullets @ 2350fps and 55gr charge:
9lb rifle - 28lbs
10lb rifle- 25.2lbs of recoil
In other words, you would have a good lion killer that recoils similarly to your 7mm Mag. You would be comfortable shooting it, you would therefore practice more and become more proficient. When it comes time to drop a Lion you will be more likely to make a solid killing shot.
Since you have a good budget you could even just pick up a CZ 550 in 9.3x62 for less than $1000. Take it to the range and start practicing. If you like it you can have it customized to your specs. If you think the recoil is light then you can sell it for close to what you bought it for and find a .375 H&H - although I doubt the .375 will, under 99% of situations, kill the lion any deader.