Unfortunately, this whole plot arc is fabricated way beyond the limits of plausible historical adaptation. There was no official NASA program to train women astronauts, just an outlaw thing put together by the head of the Lovelace Clinic. The congressional hearing didn't get anywhere near that level of press notice, Trudy Cooper didn't know any of the women in the "program" and would never have gone to Washington for the hearing.
FWIW, the Soviet's flight of Tereshkova was essentially a PR stunt, but the women cosmonauts trained for that phase of their spaceflight program did get real cosmonaut training.
Bull. The scriptwriting rule for historical adaptation is that you don't fabricate events unless needed to keep the drama going. There was plenty of real-life drama happening during the Mercury program that the writers could have used, but instead they're resorting to sloppy soap-opera tropes. It's just hack writing.
I'm not sure they said on that show that the "Mercury 13" aka FLAT were anything more than something put together by the Lovelace Clinic. It didn't go much of anywhere, though they got sufficient attention - both on the show and in real life - to warrant a Congressional hearing.
The "Dot Bingham" character is fictional anyway, so I think Trudy could fictionally know her. It would seem weird for Trudy to go to the hearing (and for nobody to mention that), but I think that's in the realm of invention in the interest of tying historical events together for dramatic purposes.
The attempt to dramatize the (at the time pretty nascent) women's movement did strike me as kind of heavy-handed in general, though. The Rene-Carpenter-as-columnist storylet didn't seem believable: a column by an astro-wife seems like it would be pretty appealing to the editor of a daily paper ... and apparently Carpenter really did write one, though I think not until a few years later. I also wonder why it wouldn't have violated the exclusive with Life? Maybe the contract expired with the last Mercury mission.
You're right, that would have violated the LIFE deal, and I believe Rene didn't start writing columns until a year or two later. The LIFE deal did extend beyond Mercury, because in the source book the Mercury wives are miffed at the New Nine wives getting cut in on a less lucrative but still plentiful extension of the deal to them.
In about 6-9 months from this point in the historical timeline Scott Carpenter gets grounded due to injury, and that might have released Rene from the LIFE deal. Or else the original terms had run out by then. It wasn't a long-term deal.