Done that route many times, Houston and back, and that time of year your 1st concern would be the RV killer,mostly summer months...and that would be Cabbage just east of Pendleton. Then, again my concern, would be I-80 from Green River to Laramie...a vast stretch of nothing. There are many areas you could have weather issues that time of year but there is almost always ways to deal with it.
But given that I prefer that route than doing the get south then east thing. So do your do diligence on Cabbage in regards to the ability of your tow vehicle.