Take 89 north. The route you describe through Kingman will add hours to your drive. And you have to go through all that traffic in Vegas.

Hiway 89 through Page Az is an easier pull. But 89A over the Kaibab is not that bad and a MUCH prettier drive. It is a narrow two lane road and climbs from 4000 feet to over 7200 feet. I see Semi trucks on it all the time. Granted most are Lumber and Sawmill trucks. Winter months it gets snowy, but we should be past most of that by now.  It has lots of places to pull off in the shade and stretch your horses. As well as a lot of wildlife, Turkeys, elk and deer. Maybe I should allow you to decide which is prettier.  Through Page is red rock country. Over the Kaibab you have some of both. About an hour of Red rock through Marble Canyon and Lee's Ferry and then Big tall Pondeosa Pine trees up around Jacobs Lake.

Either way will meet up in Kanab Utah. continue north on 89 to Panguitch. Panguitch has a fairground and I'm pretty sure they have some over night stalls if you want to stop and spend the night

About 20 miles north of Panguitch take Hiway 20.  It will cross over the mountain range and join with I-15 just south of Beaver. A lot of the Semi's heading South from Salt Lake to Phoenix use this route. Again it has a pretty good climb, but is very doable unless your truck is under powered for the trailer.

Not to mention going thru the Dam!  I HATE that area, especially pulling a trailer.  When we moved up here, I had the bright idea to go thru Laughlin since we had 2 good sized goosenecks & the U-Haul going together - thought that would be easier than dealing w/idiot tourists stepping out in front of us at the Dam....well, it was hotter than blazes & the climb up out of there dang near took out the next transmission out of my truck!  I don't have a map in front of me now, but I pulled a big loaded horsetrailer up 89 to Moab & didn't have a bit of trouble - not sure if that's the route you're considering.  Only thing is to watch your fuel...there used to be a LONG stretch w/o facilities.

Moab is on the eastern side of the state. They will be coming up through Kanab.  But it is Highway 89.

I know the dam you are talking about is the Hoover dam near Vegas.  The road up through Page Arizona and then over to Kanab goes across Glen Canyon (Lake Powell) But that is not a big deal to drive across. I worry more about the Navajo sheep & goats on the road as you cross the Indian reservation than the tourist around the dam.