Last month we were on that route from Cody to Greybull, then down to Cheyenne. US-14 was a great route pulling our 44' toy hauler. Can't speak to the stretch from Greybull on to Dayton WY though. I would suspect since it is a US designated route it would be fine.
I (and most all of the Heartlanders going from the first Wyoming Heartland Rally in Cody - 2011?) took highway US 16 Eastbound over the Bighorn mountains to the National Heartland Rally in Gillette. It was doable in spite of the 20 miles of dirt road construction with guide vehicle, going on near the 10,000 foot elevation summit then. As a newby, I learned a lot, overheated my truck transmission going up, and smoked my truck brakes going down.
I haven't taken Hwy 14, but heard a lot about it in RVers online forums. I found this terrific article on the whole Yellowstone/Cody/Grand Tetons experience online this morning. Be SURE to read the section in the middle on the Bighorn Mountains and look at the picture of the BIG Yellow Sign for RVers and truckers: http://www.rvtechmag.com/travel/4_yellowstonebyrv.php?pg=all
Read the paragraph directly under the yellow sign image saying to NEVER take an RV Westbound on Hwy 14 or 14A due to steep downgrades and NO Turnouts to cool your brakes.
Not to confuse the issue, but I believe wdk450 is actually talking about US-16 and not WY-16. I don't think the OP would want to drop down and use US-16 because it would take him way out of his way coming from Dayton WY to Cody.
I use the Garmin RV GPS to make sure I stay off roads I am not supposed to be on. It allows you to put in your length, width, height, and weight and will check that against road restrictions. The only time it let me down was when it told me I could go through a road that had a low clearance bridge in Ohio, which luckily I saw online when reviewing directions! It also kept trying to turn me down a road near Natchez Trace in TN that didn't exist. The traffic feature also does some goofy study sometimes, but for the most part, I feel like it keeps me out of trouble.
I think the worst road we have done so far in terms of steep grades has been CO-139 while driving from Vernal, UT to Thompson Springs, UT. The speed limit dropped to 25 and there were several switchbacks that required dropping down to 15. I am pulling at 44' Fifth Wheel so it required a lot of concentration going around the turns but overall I felt ok. We were lucky to have a Pickup Truck that kept traffic pretty far behind us which also helped. Even with the steep grades down, the engine/transmission did most of the braking for me, and I only had to tap on the brakes to keep the speed down or slow down for the sharp turns. My best advice is just to not let your speed get away from you so you don't have to use your brakes.