Really? People cover songs all the time without trouble
Bands that cover songs, without trouble, are the bands that take care of few things... like licensing.
Just because a band has a license to cover certain songs does not mean they also have the right to license off their own version. They *may* have that right... if that's what they negotiated & payed for with whomever they were dealing with.
It's up to you to make sure that they do, and that you have the paperwork to prove it.
You can even go directly to ASCAP & BMI and do rough calculations on what licenses you need (and how much they cost). There's more than one license, btw.
in a small film as long as I'm not stealing the original song without permission there shouldn't be any legal problems.
That's so cute.
Why are you making a film, if you are hoping that noone sees it?
If your film is going to be a small project that noone ever sees (personal demo-reel, an editing exercise, film for the 3 people in film club) then sure... odds are good that it will slide.
The minute someone starts paying real attention to it (and people are talking about it), you'll be finding out why it's better to take care of the legal details up front.
The Doors probably ain't gonna be cheap (even a cover version), but there's no shortage of musicians out there with a similar sound & similar songs who write their own material, and who would love to have their material placed in a film... and who'll give it away for free.