1....cz 550 fs in 6.5x55 it definitely fits all boxes and certainly is a "shoot all day" candidate without being vanilla.
2....cz 550 fs in 7x64 most definitely not vanilla and meets all your criteria.
3....rem mod 7 fs in 7-08. Same as the cz 6.5 and one can shoot 175 gn parts for moose or bear, again a "shoot all day" candidate.
4....rem mod 7 fs in 350 rm certainly not vanilla but shoot all day could be somewhat trying.
5....rem mod 7 ks, again in 350 rm. A sweetheart to carry, 20" barrel, kevlar stock, but again very close to your 338 so not really an in between cartridge.
6....rem mod 7 358 win.....fits all the boxes, this one is blued steel/walnut.
7....rem mod 7 in 300 saum. This is ss/syn and again fits all the wishes admirably with it's 22" barrel. Think what this one might do with 220 gn nosler parts semi sp.
8....rem 700 classic in 8x57. Not the least bit vanilla and one helluva cartridge in a modern bolt gun.........checks all the boxes.
9....zastava full stock in 7x57 20" barrel and a good fit for your stated requirements.
10....rem 700 classic in 35 whelen, pretty close to your 338 but still in between, 22" barrel keeps it fairly short and light.
11....cz 550 fs in 9.3x62 not really in between your existing rifles but short light and a real killing machine.
12....i have 2 full stock 308s, one sako av and a ruger rsi. Both check all boxes but of course are not the right flavor, being exceptionally vanilla.
13....rem mod 7 in 260. Maybe a tad on the light side cartridge wise, but still with 140 gn bullets an adequate "shoot all day" rifle/cartridge combo for your intended usage.
14....i almost forgot my mod 70 featherweight (push feed 80s model) in the macmillan stock in 280 rem, 22" barrel and light. Another "fits all the boxes"