Is it required for effectiveness to stretch interior window film?
Tightening the transparent plastic is mainly for cosmetic reasons. As long as the tape stays in place, the plastic will do its job of preventing air flow.

If the kit is exposed to buffeting winds strong enough to snap the plastic sheet back and forth, it could tear or pull off the tape. So you want it to be as tight as possible.

In general most people find that it's impossible to pull the sheet tight without leaving a few wrinkles, but after using the right color tape and shrinking the plastic a little, the result is indistinguishable from a windowpane.

In your case, with that AC in place, you're not going to get a windowpane effect anyway, so just pull it tight manually. You might consider taping both the inside and outside to help prevent loosening or blowout.

EDIT: I have just learned of another reason for tightening a window film. The plastic film can not only eliminate through drafts, but can be installed to reduce convection transfer by trapping a thin layer of air against the glass. 

The ideal separation distance between film and glass is 1/2 inch. Even 1 inch is too thick as it allows the trapped air to circulate freely within the gap. But at 1/2 inch, unless the plastic film is quite tight, it will bow in and stick to the glass over at least part of the window.

I know this remark does not apply to the OP's circumstance but I include it here for the general case.