You can find a brochure for your RV here:
https://winnebagoind.com/product-res...ct-information
That may give you the towing capacity. Probably 5000 pounds.
HOWEVER, there is also a figure for GCWR. That is the weight of your RV with everything in it ready to go RV'ing including passengers plus the weight of the truck. You need to weigh the RV at a truck scale to find your actual weight. Do a search for "CAT Scale Locations" to find a place to weigh your rig.
I think a full sized, pretty much empty, p/u is at least 5000 pounds.
BTW, don't worry about going slow up the mountains, or down for that matter. Going up you want to keep your RPM's up in the 3500 to 4500 range to keep from overheating. If that means being in 3rd gear at 35-45mph then that is the correct speed.