You could try I25 south into Colorado to exit 281. 18 miles or so I believe E CR-70/Owl Canyon Road west. That takes you to 287 & if 287 is open your home free. I believe it is about 75 miles total vs the 45 from Chey to Lar on 80.
Happy Jack Rd is almost always closed when 80 is closed. The closure gate is about 20 miles closer to Laramie though so if the roads do open you are that much closer.
There is a gravel road north of Cheyenne that is supposed to go to Lar as well but since I've never taken it I can't recommend it. I heard it can be washboarded. You take hwy 211/Horse Creek Rd to Horse Creek, then Fisher Canyon Rd/CR228 Laramie which turns into Roger Canyon Rd/CR17-Albany County & drops you in to north Laramie. Make sure you verify these roads because I haven't actually driven them.
Let me know how the 211 Horse Creek option is if you take it.
Good luck.