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In addition to pushing forward on the FOIA approach, I'd suggest trying to interest whatever group the NSA has assigned with historical preservation. I'm sure they are understaffed and underfunded but at least they have an interest in preserving things like this, whereas the FOIA people's goal is to find grounds to deny requests as cheaply and quickly as possible.

I imagine there are NSA history buffs outside the agency and contacting them first to get their input on who and how to approach (or maybe even an introduction) would probably be better than a cold approach. Two things that may help:

* Make it easy for them by doing the legwork up front. Find a preservationist or museum with a working 1-inch VTR, ideally one of the "portable" ones (actually like a large suitcase vs a washing machine). It may be that removing the tape from the NSA facility makes it a much bigger ask (keeping in mind they don't know for sure what's on it yet). The closer you can make the initial proposition sound to "it just takes one of your people to get the tape for a couple hours to digitize it. We'll set it up but NOT be in the room for the playback, then NSA can decide what to do with it" the more likely success will be. (obviously, it'll probably be more complicated but once they're in...)

* Highlight that this is ONLY about preservation not release. Old magnetic tapes degrade with time. Accessing playback gear may not be possible in another decade. It's probably still savable this year but will soon be lost forever (create urgency with FOMO).

Don't even talk about release or FOIA, make it about some NSA history initiative that already has support. Ideally, they're working on a "Women in NSA" or "NSA and Early Computing" exhibit for next year. Just between us though, once you know that the video exists digitally, the grounds for FOIA denial is gone :-)

Good luck!