Why do you think they would need to do those launches in quick succession? If the lunar lander starship is parked in orbit without a crew it can be refuelled over any convenient period of time. Once it's fuelled just send up the crew in a dragon capsule and off you go. The dragon would probably make a good lifeboat if you brought it along for the ride somehow.
Also it does not need to be the same tanker to do the resupply. The plan is to make 2 starships per week. There should be plenty available to continue refuelling missions if one is not available.
Wikipedia[1] cites a 2010 study that gives a boiloff rate of 0.1% per day for hydrolox. Not sure if it would be less for pure LOX. Methane would remain stable so I wonder if there's a way you could balance the fuel load so that you can deliver only methane on some of the trips and then all the LOX in one trip to avoid having it hang around in orbit.
Oxygen is the vast majority of the propellant mass, I'm afraid. Methalox is CH4+2O2->CO2+2H2O, so you need 4 Oxygen atoms for every Carbon atom (with Hydrogen being a rounding error).
As long as the crewed starship is fuelled in orbit not too long after the last tanker ship launched, then the amount of boiloff will be negligible.
I suspect Spacex will develop cooling systems to re-liquefy any gaseous propellants as this will be more of a concern for the ~6 month mission to Mars.
Dragon is 4.4 meters (14.4 feet) tall and 3.66 meters (12 feet) in diameter. Starship fairing payload envelope is 22m tall and 8m in diameter. More than 3 dragons would comfortably fit inside.
I think the idea is that the fueled-up Starship in LEO would have already have most the payload inside since it launched. The only thing it would be lacking is the astronauts, so they can come in a much smaller vehicle.
That said, this contract has astronauts boarding Starship in lunar orbit, not Earth orbit.
