I'm glad that Country Coach owners are now replying as it's better to get advice from somebody who knows your coach (unlike me who has never owned a CC).
Yes, as you may know, there was quite a fiasco during that time period from the mid-1990s into the early 2000s where Toyo tires were failing on Country Coach motorhomes as Country Coach used Toyo as OEM back then.
It was ultimately determined (and this is from my recollection from those days) that Country Coach was determined to be at fault as they didn't properly determine the correct size and ratings of the tires they were putting on their coaches that were leaving their Oregon factory. They were using improper tire sizes and ratings as they were not calculating the correct GAWR of their coaches. From what I recall, Toyo was not liable and the tire failures were not because of defective tires but rather CC using the incorrect tire sizes and ratings that they were specifying as OEM and consequently, many of the failures were resulting because of both overloaded and underinflated tires. No matter what tire manufacturer CC used at the time, the same same failures would most likely have happened.
Toyo from that point on (somewhere around 2006 when the lawsuits were settled), refused to label, suggest, advertise or otherwise market any of their tires as a specific "RV tire." That's one of the reasons you don't see that to this day ...i.e. Toyo advertising or marketing any of their tires as "RV tires" as Michelin and Goodyear do. In turn, CC stopped using Toyo as OEM at about the same time.
The reason I mention this is that it seems the key year here is 2006 --the year of your coach. Again, from my recollection --and Country Coach owners can correct me if wrong-- 2006 was the "transition year" where CC made the decision to use Goodyear instead of Toyo for OEM. Therefore, from my memory, they used both the 12R22.5 and the 295/80R22.5 as the transition was being made.
So I still don't understand why you ended up with 11R22.5 as that does seem like it would be undersized to me.
List your model and model designation and let's see what those who have the exact coach as you do have used or at least by CC owners who have and will further responded in this thread are recommending. I'm guessing, yes, go with a 295/80R22.5 (or alternatively a 12R22.5)
Brand is less important. If you like Hankook, so do I as it's one of the brands in my top 4 or 5 brands, then consider them. I actually do not see a Hankook in a 295/80R22.5 but they do have the AH37 in a 12R22.5 But the other brands I tend to like are listed below.
As is being said recently on the forum, many of us, myself included, tend to overthink the brand of tire when considering new tires. Any of the well-known brands will serve you well and will be safe and reliable. As many know, I personally do not like tires that are coming from corporations that are based in the PRC (China) but that's a personal thing with me. The ones listed below are coming from companies that ultimately have their global headquarters in South Korea or Japan. Also, Sumitomo and Yokohama are now making many of their commercial truck tires in the U.S.
Hankook AH37 12R22.5 H/16PR
Toyo M 144 295/80R22.5 H/16PR
Falken RI151 (a Sumitomo brand) 295/80R22.5 H/16PR
Yokohama 104ZR 295/80R22.5 H/16PR