For a deer rifle I'd pick either of those calibers over a 7.63x39. All 3 will kill deer, but the x39 is more limited by range than the other two.
The 7mm08 is a better option if anything bigger is also on the list. 243win has less recoil, could be a factor depending how big your wife is and how she handles recoil.
In my experience 7mm08 ammo is less available than 243win. I'd probably go 243win if I was buying factory ammo, 7mm08 if I was reloading - in fact I'd also get a 270win over a 7mm08 if I was buying factory ammo. My primary hunting rifle is a 7mm08, I think its an absolutely FANTASTIC cartridge, but factory ammo availability, especially in a ####ty situation like you forgot your ammo at home and have to find some quick, just isn't that great for it. Practically anywhere (gas stations, hardware stores, etc) sells 243win ammo if they sell ammo at all.
Edit to add- if you're looking at the Axis, have a look at Reliable Gun's website. They have the Axis 2 on sale right now, add in the $75usd rebate available til the end of the month and you can have an Axis 2 (which has a MUCH better trigger) for about $400 all said and done. They also have a Ruger American in 7mm08 for $600 right now.