However, it would have been a bit of an issue if the Mercury-Redstone 3 flight had fritzed in the same way the Mercury Redstone 2 flight had. It's all very well getting the first man into space, but if he drowns before being recovered because something went wrong, that's not going to do anything good for your credibility.
A lot wasn't going to happen on the manned flight. MR-2 being off course caused the LOX to deplete early than planned which fired the escape rockets making the flight higher and throwing off the recovery area. That wouldn't have happened on a the manned flight. Further (as argued by Sheppard among others) the problems were easily identifed and fixed and there was a delay getting the MR-BD (https://en.wikipedia.org/wiki/Mercury-Redstone_BD) booster delivered and ready when the booster for MR3 was already at the Cape and checked out prior to the launch of MR-2. The fixes WERE easy and already applied to the MR-3 booster before the MR-BD booster so while Von Braun insisted that there had to be a 'perfect' flight before he'd sign off on man-rating the Redstone almost everyone ELSE disagreed with him. And for once, (and pretty much based on this fact alone) he refused to back-down or go with the group consensus and arguably it cost the US the 'first'. Going back to Ham's flight it should be kept in mind that BECAUSE it wasn't a manned flight there were fewer recovery ships and they'd have moved a LOT faster to get to a manned flight than a chimp. Meanwhile the capsule leaked yes it wasn't really in danger of going down and unlike Ham the chimp Sheppard had both the means and training to get out of the vehicle if it looked to be in real danger.
The only difference that you have to match, and one up what the Russians were doing in the '60s
That's arguable depending on the circumstances. Had Kennedy not supported going ahead with the Bay of Pigs invasion, (and it was a near run thing) or it had aborted as was suggested several times then there's less pressure to 'match' the Russians. And again an American 'first' is going to totally derail that pressure since in just a few years the US will be able to out-match the Soviet lift capability with the Saturn-1 and once they have either TTL-Gemini equivalent or Apollo the 'race' is pretty much a dead even thing.
The Soviet Manned lunar program was a mess Lack of budget, lack of resources, lack of needed Technology, lack of interest of politburo, rivalry between the rocket designers etc. That all let to down fall of program and in this scenario, once the KGB confirmed that USA bury there Lunar plans, the Politburo will terminate the L3 Complex (aka cosmonaut Lunar landing) and put all effort in L1 Complex, a manned Soyuz capsule around the Moon and Almaz + TKS space station for Military (analog of MOL) So end of 1960s the Soviet made nasty surprise by sending a Human around the Moon, but that's all, no orbit, no landing.
The Soviet program was pretty much initially based on grabbing all the 'low-hanging-fruit' firsts they could get with no real rhyme or reason towards an integrated program. They had the initial advantage in both lift capacity and larger spacecraft due to the design of Vostok being bigger than the Mercury spacecraft. Hence they could eventually put two and three people into the vehicle with 'nominal' risk whereas the American's were dead-ended by the Mercury design at one and only one astronaut. The US also couldn't lift anything bigger than Mercury initially without some sort of upper stage for the Atlas which was questionable at the point where Mercury was being designed.
The USA will answer that by sending a modified Gemini around the moon with help of Titan IIIC and two Transstage
But that until soviets terminate there L1 Complex after hand full of flights...
That makes the assumption that a TTL-Gemini actually flies on Titan at all which isn't a given. Initially the Air Force was not interested in providing Titan's for launch purposes at all, and even after they did agree to provide them they had to be ordered to 'fix' a number of issues with the LV before NASA would accept them. Given any other circumstances it may very well be that NASA wouldn't use the Titan and would instead use the Saturn 1 which had more capability and four straight successful flights before the first Titan GLV flight success. Part of the reason this didn't happen OTL was because the extra capability of the Saturn as a Launch Vehicle was so superior to the Titan that it was considered a huge waste using "only" the Gemini since it could carry the proposed Apollo capsule and more into orbit on every flight. So re-winding a bit an alt-Gemini might have been a much more capable vehicle under the circumstances of a "no-Lunar-goal" scenario. Gemini OTL suffered pretty much as much as Mercury did in being required to do to much with a smaller than optimum spacecraft due to launch vehicle constraints which themselves were driven by the need to push for the 'end-goal' of going to the Moon with Apollo.
A more capable Gemini like vehicle and the Saturn 1 means should the US get a hint of Soviet intentions they might just push a TTL-Gemini into a Lunar flyby before the Russians can, again scaling the 'race' back as the Soviets would need to make a decision on continuing to push the envelope or not.
Also keep in mind that both sides had made additional plans for going to the Moon, (eventually) with what they HAD if need be rather than what they wished they had.
Would interesting scenario for a TL were Apollo and L3 Complex are abandon programs and allot of soviet lunar Lunokhod and US Prospector rovers on the Moon. Gemini and MOL still fly and while soviet use Alma and TKS
Most powerful rockets in world are the Titan IIIM/F and Proton rocket
Both MOL and Titan were very much NOT what NASA wanted and frankly given other circumstances in not having to be so focused on the Moon I have great doubts that either would have been even considered for NASA development. They just weren't what NASA needed despite what the Air Force wanted. MORL on Saturn 1 was vastly more capable and economical than Titan/MOL. Again the limiting factor of Gemini for size and capability was the Titan LV and given Saturn as an alternative it wouldn't make a lot of sense to NOT go with a more capable alt-Gemini and early MORL deployment. The main limiting factor is again Von Braun's (and the US government's in general) conservatism but give the Soviets similar to OTL 'firsts' still happening, (first two, then three man crew and multiple flights at once plus duration records) and the US would likely push forward with development of the alt-Gemini or early Apollo-like, (someone removing Max Faget's hand on the scale towards NAA probably means someone with more capability gets the actual TTL Apollo nod) spacecraft and the US may have several 'firsts' well ahead of the Soviets.
Especially if the Soviets DO fly two separate vehicles in Soyuz and TKS. If it's clear that the Soviet's have both a military and "civilian" (they didn't make any such distinction but the US might well see it that way) space flight system they US might follow suit. But again it's not a given that the alt-Gemini is designed around Titan as per OTL when NASA has no real reason to use it and has Saturn available and ready to go.
Randy