OK I just wanted to make sure that its not another manufacture that might have produced this. The Tri Glide I believe went into agreements with Leman Trikes in 2008 to build this Trike, Leman Trikes builds conversion kits for other manufactures, Leman Trikes is out of Spearfish South Dakota.
As you know that this trike is based off the Electra Glide Ultra Classic which is the conventional two wheel touring motorcycle, I'm assuming that your salesman would explained all of this to you.
So Harley Davidson supplied their parts or half the bike minus the back and then that is shipped off to Spearfish originally, after the owner of Leman passed 2012, then the assembly moved to New York, how much this is 100% Harley today is something that I'm not sure of ?
The problem you are having is something that would concern me especially for the uncomfortable fact of what the problem produces, and then getting so hot to the point that's its creating a safety issues.
Initially its telling me that the exhaust is getting so hot that I would assume its a fuel injection issue, not sure where the catalytic converter is located on this exhaust but its likely something going on there, and that can be related to the fuel management.
My suggestion to you, is get the bike back to the dealership where it was purchased and explain your disappointment and discomfort with the heat that the exhaust produces, and let them know that you feel is an issue for you, and that this is a safety issue not only for you and your passenger for the surroundings that come in contact with the exhaust. With the mileage that is on the bike and the year this has to be covered under normal warranty coverage (giving there is a defect) and if for some reason the base warranty has expired, you have an emissions warranty that is most definitely covered under.
So something is going on where it producing excessive heat into the exhaust that needs to be looked into, I don't think that I would except the answer that you might get from them and that is "this is a normal condition" for this model, if that happens it might be worth your time to call Harley Davidsons consumer services hotline and portray your disappointment in your recent purchase, at least if there is a problem down the road its on record with them.
(there could be an inherent design issue when this was designed)
Let me know of this help or you have any other concerns, I worked for a manufacture for 25 years so I have a pretty good knowledge of the internal workings and this will be in your best interest.