Well to the best of my knowledge all Military Style rifles in 308 (and the old M-1 Garand in 30-06) are fairly heavy with the exception being the new FN SCAR which costs a LOT.
The M1A is not too bad, but get heavy if you scope it.
The DPMS "AR-10 style" rifles come in calibers from 243 to 338 Federal, but I have owned one myself in 308 and I have a neighbor that had one in 243 and neither own was reliable. In fact his never once got through a mag of ammo without at least 2 jams. My 308 also seldom would go through a mag without at least one jam
I have an FN FAL that is 100% trouble free, but again it’s heavy and a small lady would not like it in all probability.
The FNAR in 308 is a Browning Short Track in 308 with a 20 round box mag but the stock is quite long even in its shortest configuration. It's lighter than the FAL, but scoped and loaded it's still a 1.4 pound rifle.
In military style rifles that would be good elk guns the ones I know of are:
The M-1 Garand,
The HK-91 and clones (CETME and so on)
The FN- FAL
The FN SCAR
The M1A and M-14s
The AR-10 and "clones"
The Dragunov sniper rifle clones in 7.62X54R. This one is one of the lighter full power autos, but pretty long.
There may be a few others that pop up now and then too. But this list constitutes most of the ones you could choose from.
What state are you in Nim Rod?