270 is an also ran, wow!!!
All rounds mentioned will work, esp. for Deer at normal ranges & can be used for Antelope. I like the 280 alot, but you did not ask about it, plus you have a 30-06 barrel & that means your upper level animals bigger than Deer are covered. The 223 takes care of
volume shooting & most varmits. So, I guess this leaves some of the long range, windy day Varmits & Deer & Antelope with bigger
animals shot with the 06.
To me this is an easy choice. The 7mm-08 is a great Deer round, but the other shooting that I see done with this barrel makes the
25-06 the clear winner. Anyone who actually hunt Antelope know that the 25-06 is more than up to the task & in fact many argue
that this round is the best available for this purpose. I have witnessed Antelope being shot with 25-06 & AI, 270, 30-06, 300WM, 30-30AI & others & the 25-06 gives up nothing for Antelope or Mule Deer to at least 400 yards.
Also, the 25-06 comes with a 26" barrel of decent weight/contour which makes it easier to shoot in the field & the 26" will help you
utilize the velocity potential of the round. This gives the round in the Handi more of an edge over the 7-08 than if they had the same barrel. As far as recoil is concerned, both rounds you asked about are mild for Big Game shooting, & for varmits the 25 would be a little less, due to lighter bullets used in this high volume application. The recoil of the 25-06 in the Deer loads is way less than the 30-06 with the powder volume being the same, but with lighter bullets, just a matter of physics.
Later on you could buy a 45-70 barrel for big, nasty stuff or just fun shooting. With those 4 barrels you would have no gaps for ANY
animals that you care to hunt.