There is one (reasonable) flat route to the east entrance from SD, and that would be to take I-90 to Billings (actually, just past it to Laurel) and take Hwy 212/310 towards Lovell/Powell/Cody, WY. (You won't go all the way to Lovell, but you will go through Powell and on through Cody.) It's a good road and flat as a pancake (almost). You will have a mountain to climb once you enter the east gate, however. Really, it shouldn't be a problem, and it's certainly easier to navigate than the road south of Gardiner.
My personal recommendation would be to exit I-90 at Buffalo, WY and take Hwy 16 over the Bighorns to Ten Sleep, then on to Worland, Greybull and Cody. Yes, Hwy 16 from Buffalo to Ten Sleep is mostly in the mountains, but it's a good, wide road with gentle climbs and descents and mostly gentle curves. There are two "hairpin" curves on the west slope, just as you enter into Ten Sleep Canyon. Recommended speed is 20 or 25 mph around each them. Then it's back to gentle curves until you reach the bottom. Unless you have a severe fear of mountain roads, this would likely be the most pleasant drive for you and one of the prettiest. (And if you DO have a severe fear of mountain roads, you probably shouldn't visit Yellowstone.)
Do not "ride" you brakes going down any long descent. Use a lower gear, and if that's not enough, then apply moderate brake pressure long enough to slow to your target speed and get your foot off the brake until you must repeat. There are ample pull-offs along the descent, and if you're using your brakes much, it would be smart to pull off and check your wheels to see if they're hot. If they are, take a lunch break of 30 minutes or more. Take some photos, get some exercise. It's a pretty place to spend some time anyway. I've driven that road a hundred times and never overheated my brakes, but it can happen to the unwary. (The descent into the canyon begins just after crossing Ten Sleep Creek, which is just past Deer Haven Lodge. The steepest portion is about a mile long, but it's not scary steep.
You won't have any more mountains from the base (a few miles east of Ten Sleep) until you're in Yellowstone.
FYI, Hwy 14 from Ranchester to Greybull has steeper climbs and descents (and 14A from the top to Lovell is very steep). Also, Hwy 212 from Red Lodge to the NE gate is quite steep and curvy, and I would definitely not recommend it to a flatlander towing a heavy trailer.
Actually, none of these roads should give anyone a problem who is careful and doesn't overheat the brakes. The only road I've mentioned that's a bit gnarly would be the one in the park south of Gardiner. It's a bit narrow and curvy for a mile or two, but 100s of thousands of tourists take it every year, many in their big motorhomes or TTs.
Oh, and I agree with Jim Shoe that if you want to avoid all mountain driving while towing, take his suggest route.