I think your husbandry might be a bit off.
If you are using tape, what brand?
I don't think you'll find anyone here using heat tape, at least not the serious keepers.
Chams are a bit different from "heat lizards".
You are correct. Chameleons are different to most lizards. Besides the obvious physical characteristics, they are also arboreal creatures. The live in trees, whereas the "heat lizards" I think you are referring to (Bearded Dragons, etc?) are mainly ground dwelling.
However, being reptiles they have similar needs when it comes to warmth. Any diurnal reptile needs to warm up its body in order to function. They need heat not only to be able to hunt, but also to be able to digest the food they have caught.
And whilst it is true that certain species of chams don't need as much heat as others, most of the bigger species (including Jackson's) do in fact bask. They don't go and lie out on a hot rock like some of the "heat lizards", but they do climb to the upper parts of trees in order to find direct sunlight to heat up. The amount of direct sunlight, and the time that they spend in it will be determined by their heat requirements.
In captivity, keepers mimic the way the sun provides heat to chameleons by providing a heat lamp under which the cham can bask.
Nowhere in nature do arboreal chameleons derive their heat from sitting on warm perches. They don't ever walk on hot rocks, and I don't know of natural tree branches that would heat up in the way you describe your device would. And the main aim of chameleon husbandry is to mimic their natural environment as best as is practical.
My Jackson's likes to bask under the UV thinking that he is going to get warm (he doesn't).
When you say, he likes to bask under the UV lighting: if you mean he sits comfortably for most of the day on a perch under the UV light, it's more than likely that he IS getting warm there. A chameleon NEEDS the heat - he wouldn't be comfortable until he got it (unless you have some temp guns monitoring him, I don't know how you can tell he isn't getting warm enough).
If on the other hand, he is climbing up the side of the cage, or up onto the top of the cage, hanging upside down, trying to get closer to the UV lamp to bask, then he ISN'T finding anywhere to get warm enough, and he is not basking at all, but desperately trying to find a way to get closer to some source of warmth. In that case, you need to provide a heat lamp to create a basking spot.
If your primary concern is that he is spending too much time under a UV light, and not enough under a heat lamp that you set up, then it sounds like you have a problem with your cage design.
You should design a cage so that there is both a UV and a heat gradient, and that a chameleon can spend time basking where he can choose to be exposed to UV or not. In other words, you should have your lights set up so that the cham can get warm, AND still be in range of the UV light.
and 2) great fear of burns and/or fire.
Again, this is a cage design issue. If your cage is designed and set up properly, even with a heat lamp there should be no burns or fire. There is plenty of information on this site, as well as other good chameleon websites that will explain how to set up heat lights safely for you and your cham.
(Also I think that heat tape in a wet environment - as a Jackson's cage should be - would probably be more of a hazard).
Like a wise member on this site once said to me (Zerah, you know who you are

): you should first rethink your husbandry before going off to buy the latest gadgety-type reptile product.
Damn, my slow internet connection allowed you to beat me to the reply, Zerah