It takes an enormous amount of power to create heat. You would need a rather big battery to be practical I think.
True, but you only need the heat for a few seconds at time to shrink tubing. The link to the one such heat gun I found shows that it's practical, just not yet very affordable.
Nah I think it's priced that way cause there's no competition......yet.
I gave you ~1/2 hr yesterday looking for one and I must say your Weller looks a cool bit of gear.
But I'd keep an eye on what Dewalt, Milwaukee, Makita and Hitachi are offering in their cordless ranges along with a detailed email to their head offices explaining your exact requirements.
Even 2 years ago we didn't have the vast range of cordless gear like we do now and you can be sure there'll be others that have a use case like yours, something mightn't be far from release.