Which Lee-Enfield? Sort of matters what Mosin you have too. That's more about it being a rifle or carbine.
The cartridge isn't the question. It's which rifle shoots better. However, you'll have to slug the barrel of the Enfield to determine who's ammo to use(you'd need to slug for reloading too). Not all Enfield barrels are .311". And commercial hunting ammo uses either .311" or .312" bullets.
Cabela's(who will send anything you buy on-line to your local Cabela's shop) and Midway both carry 7.62 x 54R and .303 Brit with SP. 150(more than enough for deer) and 180's in the Brit. 180's in the Russianicon.
It's May. You have lots of time to buy a box of each, try 'em and sight in.
Oh and I did the Wal-Mart Test online. They don't know what either cartridge is.