At the velocity that the 7-08 pushes a 140gr bullet, you would get along just fine with a conventional design bullet. I've got a sweet load worked up for my father's Marlin XS7 in 7-08 that uses a 139gr Horn BTIL at 2755fps over top of 41.5gr Varget and Fed210 primers in WW brass. You wouldn't beleive me if I told you the group sizes that we're getting with this load, but let's just say that quite often you can cover 3 shots with a loonie at 200 yards. If you really wanted to maximize penetration, I sure do love the 140gr TTSX, but I mainly use it in my .280 and 7RM where the velocity is a little more demanding of a high-integrity bullet.
I would either use the 139gr Horn or else if you're going to run a 120gr bullet, I would run the TTSX since it'll be going a fair bit faster than the 140gr group of bullets. I must say, though, the 120gr BT is a VERY tough version of the BT, and I've seen many a picture of moose that were made very dead by that bullet out of the 7-08. You may want to develop a load with the 120 BT as well as the 120 TTSX and if you can get them to his the same POI you could use the BT for practice and deer, and then the TTSX for moose.
For powders in the 7-08, Varget and RL-15 seem to be the most popular powders that are used with great success. Varget gives superior accuracy over everything else in my father's rifle, so we use that. Many guys say the same about RL-15.