If one has the Saturn V production line open, why not use a Saturn INT variant for LEO ops?
The short version is that the Saturn LRB is absolutely the project lead in this timeline.
Most scenarios I can come up with that use INT (and don't completely change the funding environment in the 70s and 80s) end up with INT-20 and no Saturn V. The concept here is a plausible scenario under which the SV sticks around. The other path I can see this taking reasonably would be to go ahead with Apollo/Titan for LEO, but for that to make sense you need to justify Congress going for both a substantial lunar and LEO program simultaneously in the back half of the 70s, which I have a hard time picturing. The high concept is that Nixon preserves lunar exploration by making Saturn V THE NASA vehicle and the late 70s gets an inversion of the inability to do BLEO missions under STS.
The way I see this playing out is that the Saturn IB fleet is stretched (by producing a couple extra SIVB stages) through Skylab, ASTP, ASTP 2 (Skylab/Salyut, enabled by there being no Saturn V shortage) and a leo test program for Moonlab, with no clear LEO plan beyond that. By which point a few extended duration LM Hab lunar surface missions have flown and a LESA like base is the project for the 80s. The combined Booster/C-3 program looks like a way to piggyback a form of LEO access on Saturn V upgrades for lunar base construction when it becomes clear a shuttle isn't in the cards without giving up on the Moon (with the nice benefit that this twin F-1 booster looks like a very nice platform for unmanned deep space). The equivalent to the shuttle design evolution in this timeline being a fight between thge enhanced Saturn V with a 10m SIV stage lunar base advocates want and a common core Saturn using the dual F-1 260" core that looks like a cost reduction strategy OMB would approve of (how much of a hammerhead do you think a 260" dual F-1 stage with four common core boosters handle?)
I tend to think that Common Core would look very attractive for a while but run into serious cost issues when the reality of redesigning for S-II for increased payload with reduced diameter (and no M-1 engine) hits.
Bear in mind that none of this is a true C-3, in that I'm using the 260" diameter proposed for Saturn V boosters and used on the SIVB rather than the 8.8m that was planned earlier. In other words, it much more closely resembles the 1980s Jarvis proposal than something out of the early 60s.
PS: I didn't mention the Saturn II option for LEO, and frankly my thoughts on that boils down to two threads. On the one hand I don't think there is money for HG-3, and on the other I think that a Saturn II that needs SRB's basically defeats the purpose, given the S-II's price. The bigger issue with this TL IMO is ARJ260 and INT-05 (the Constellation architecture in 1981 anyone?). At the end of the day I don't have a good answer for why an F-1 booster would be the path chosen, frankly all else being equal it wouldn't be. The best answer I can come up with for the moment is common core, and this sort of issue is one of the big reasons this is still something I'm playing with rather than a TL I'm trying to actually write.