I was thinking about this earlier today. The real difficulty is that feminism wasn't quite where it was later on--the Russians were Stone-Age, flying Tereshkova was just a propoganda stunt, the cosmonauts were super...well...dickish. For example, during Tereshkova's flight, she was supposed to have the callsign "Golden Eagle". The cosmonauts didn't like this, so she got reassigned the callsign "Seagull". And of course, the Soviets* only ever flew two female cosmonauts, both of them for pure propaganda (the second flew right before the US was going to fly women on the Shuttle). So having the Russians fly more female cosmonauts probably isn't going to happen without the US flying female astronauts, and probably not even then. That wouldn't get them any prestige, or one-up the Americans, but that's by far the easiest way for the US to accept the FLATs, especially as feminism burgeons throughout the decade. If the Soviets launch a couple of women, the US will probably launch at least one, otherwise they'll start to look bad.
And it's a bit tricky to see Congress or the NASA administration (and especially the astronaut corps and Deke Slayton) just decide to accept female astronauts. Thus, I think the best way to get the FLATs into space is to advance feminism by about a decade, which then runs into other problems: this means that the Betty Friedman stage has to happen about 1950, which is a little too early IMHO. That was the early days of the "return to normality" from WWII, where women were being forced back into the home, so the repressed suburban lifestyle wasn't quite all there yet. Plus, for the more radical second-wave to kick off, the preceding civil rights, Vietnam War, and student protest movements are really helpful. While certainly a pretty decent proportion of women would be able to devise the tactics used there on their own, without having participated in earlier movements I suspect fewer of them would be as radicalized or as educated in the ways of protest as they actually were (nor, for the younger college-educated and wanting to use it set, as pissed off about their marginalization in the protest movements--seriously, that was a big-time boys club).
Once you get feminism in place, it's fairly likely Congress will move to start accepting women into a number of previously "male-only" things in 5-10 years, so you could just squeak the FLATs through by '65 or so, or have some faces that never made it into the history books IOTL (too old for Group 8, too young for the FLATs) end up in the program, along probably with the first wave of black candidates (like Robert Lawrence IOTL--it's really a shame he died in that crash). More or less Group 8 (a few females and blacks, maybe some civilian scientists, but still mostly white male test pilots), except about a decade, decade and a half earlier.
Of course, a social change that big would completely alter the entire political landscape, not excepting the space program. So it's a tricky thing, getting the FLATs in.
EDIT: Of course, in the longer run and somewhat ironically, women are better suited for spaceflight than men. They're on average a bit smaller and less resource consuming (saves on weight--always important in spaceflight!), somewhat less susceptible to space adaptation syndrome, and perform better psychologically in crowded conditions (like those found on space stations). They're not huge differences, but they're there.
Another way to do it might be for Jackie Cochran (who had organized the original tests) to not withdraw her support right as the FLATs were actually starting to have to fight. This was important, because she was one of the most influential backers (due to her connections) of the entire project. Have the Russians do a bit worse, or the Americans a bit better, so that the US appears to be catching up or at least not falling behind. That might just be enough for the FLATs to fly one or two flights, at least, before the leap to Shuttle.
*I got this slightly wrong earlier. The Russians did fly a third female cosmonaut, Yelena Kondakova, in the '90s. I hadn't heard of her before, because I know much more about the Soviet than the post-Soviet program (to be entirely honest).