I drive Monteagle both ways frequently. Just ran it in both directions with our gasser MH towing a dolly (+/-20,000 GCVWR) had plenty of power and never dropped below 50 on the uphill(and had extra pedal), held that on the downhill with minimal braking.
Last year we made the run often with our 12000 K lbs fiver pulled by our F250 gasser. It was no problem and maintained speed with minimal downshifts going up and held very well headed down.
3 year ago we made the run frequently pulling a 27 foot travel trailer with a Yukon. That one worked the Yukon hard until I added a tranny cooler then it was OK on the trips up and down.
Frankly admit early on I was intimidated a bit by Monteagle, but after running to Alaska and back from Georgia, Monteagle really is now easy and no sweat.I found my worry to be overblown and I have been glad I useI-24.
I do avoid ATL when northbound from where I live or Florida by coming up from Tifton on the 520, through Columbus and to LaGrange on I-185 then on to US 27 in LaGrange to Rome and GA 1. From Rome on GA 1 I connect with GA 53 to I-75 at Calhoun. That route is 4 lane all the way 55-65- minimal traffic and not a lot of stops (a few in LaGrange and Carrolton but without heavy traffic). Time wise its the same as tackling ATL considering ATL traffic- do it about 5 times a year and have none of the Atlanta hassle and tension.
Personally I avoid I 65 in Alabama as our rigs have been beat to death with the condition of the road surface and it takes me a bit longer, but others may see it differently.
I assume your tow vehicle is OK for the rig you tow. I think you'll do just fine. There is a rest area in both directions at the top of the mountain to take a break.