Admittedly there should be an easier way to handle breaking down doors, especially because of how impossible navigating through the military tower is without expert Technical skill. Of course, doing so shouldn't be without consequence: maybe it could attract the attention of aliens who were otherwise minding their own business.
Alternatively, the security and military towers don't have to be nearly as inaccessible once a certain element is added to those towers. You'll still end up not going there unless you're mega-prepared or desperate.
Unfortunately, I used up all my security keys, so unless there's a way to fix elevators...
There's a way to fix elevators. Unfortunately in your case, that also requires at least an advanced Technical skill, not to mention at most five whole multi-tools.
I ended up scouring all of the towers and actually found two more security keys! :) Then I went back upstairs, went through a couple of doors... and found myself in the Security tower, blocked by another security door! Argh! :P
Well that's just your own fault. Military tower has exits to the Security and Storage towers, the latter of which won't need any keys. The Wiki has a small diagram showing the whole layout.