The problem with this, is that even at the beginning the USSR had serious problems at home:
- All their early firsts had been (relatively) inexpensive, with a high PR Value, so securing support for it wasn't terribly difficult
- The Soviet Military pretty much held the purse strings, so anything relating to rockets needed their support, and thus practical military benefits were a major need
- The R-7 ICBM, while pretty useless as a military tool, was somewhat easy to uprate for Soyuz-sized payloads by kludging in new upper stages, given its sheer overdesign in its early days
- The consequence of this is that larger LVs were something they didn't have experience in, IIRC the UR-500 wasn't approved until 1962
- So trying to build the 75,000 Kg (later uprated to 95,000 Kg) to LEO N1 was a massive leap, one too much for them
- A late start of 1964 vs 1961
- Korolev dying on the operating table
- Add to this they're only having about 10% of the funding NASA had for Apollo/Saturn, and you begin to see where everything began to fall apart
For me, the only viable POD is 1961, by having the USSR take the US Commitment to Manned Lunar Landing & Return seriously.
From here, there are two basic paths they can take insofar as Launch Vehicles are concerned:
1 - N1. From Korolev's OKB-1. The one that was (eventually) selected IOTL, Three LOX/Kerosene stages to put the Manned Lunar Complex into LEO, either all in one go, or through two or more launches.
2 - UR-700. Chelomei's OKB-52. The only real competitor to the N1 in terms of capability. 151,000 Kg to LEO, utilising a Direct Launch Profile, though relied on UMDH/N2O4 propellant mix. Storable, but toxic.
That said, whichever is chosen will face the same issue, mainly that the scale of such an undertaking would prove too much for just one group to handle. Something Yangel realised far sooner, when he proposed having his, Chelomei's and Korolev's three agencies pool their resources into the effort, by focusing on one specific aspect of the mission. Incidentally, Yangel's Bureau did develop the LK Lander IOTL.
So let us say that OKB-1 develops the N1, but the L3 element is given to OKB-52. Now you have reduced pressure on both groups, and a requirement to cooperate from the Central Command. From here, they now have a realistic chance of achieving a Manned Lunar Landing at all.
That said, the 4th of July 1969 is a poor date for landing IIRC, given Orbital Mechanics means that too much of the Moon's visible side is in night. A day or so ahead of the US? Doable, though still unlikely.
Such a tight margin, would, IMO, be a solid incentive to keep going. At the very least, I can see LESA happening, under the maxim of "We're not going to the Moon to visit. We're going to the Moon to Stay!"
And that's one area where the US's sheer economic power would assure them, IMHO.