Short answer: No
Long answer: Yes, the transclusion widget sets a variable called
"transclusion" that (aside some other things) also includes the name of
the template being used (I am guessing that it would in most cases). So
if you can write javascript, you may be able to devise a macro that
extracts the name of the template used that you would then be able to
use to get the template's fields. I can't think of any other way right now.
Of course, a better way of permanently making this sort of thing easier
would be to change the tiddler-widget to set an appropriate variable
containing that information. Maybe you can open a github issue about this ?
/Andreas