Honestly, the best way to rank up your access with any engineer is to just keep modding stuff. Doesn't matter what it is, even something you don't really need, just spend the resources and mod it and your access rank will increase. The trick then is to always bring a lot more engineering mats with than you think you'll actually need, so you can rank any newly unlocked engineer straight up to grade 5 access with your first visit. Means gathering more resources before you visit, but that's fine, because you'll need to use them eventually anyway. Sometimes a good way to do it is to continually engineering the same module at a lower level, until you fill the entire circle on the module, just to gain the increased access, before engineering the higher grade on that module, so by the time you get to modding at grade 4, it only takes a couple of rolls and you're at max access level.
Oh and on the whole "log out HGE" "exploit", maybe I'm in the wrong, but, is it really any different from people running laps around Davs Hope? Or logging out and in to reset a Guardian Ruin? You wouldn't expect people to fly all the way to Guardian space to land on a planet, go through all the pylon raising, get the blueprint, then fly all the way back to cash in and get a module, to then fly all the way back and do it again, would you? Worse if they need 2 BPs for a turreted weapon and don't even get anything from the first journey back! Of course not, they'd fly there, land, do the puzzle, log out, reset the site, and then run it again. Is doing that also an exploit, or just good common sense, or are you really supposed to do each 1 one at a time? If you had to do each BP one at a time the long way like that, I doubt most people would ever bother doing any more than once.
Is it different with those because they're a fixed site, not a transient instance that Fdev only want you to be able to access once? I don't really see the difference. You still have to track down HGEs on the galmap, FSS the system to find them, and do a lot more work logging back out of the game and loop the looping back into the HGE zone to get the goods than you do with say, Jameson Crash Site, where you literally just sit still for 30 minutes doing less work than even a Guardian Site would require. So should every zone like this only ever yield rewards one time only to prevent any kind of log out shenanigans? Hah, good luck ever getting anything unlocked ever again if that's the case! Is it more a problem because these sites are giving us Grade 5 stuff? Maybe that's the issue?
Even doing things this way though, I can tell you, it still takes an absolute age, so I can only imagine the hundreds of hours it would take to acquire decent amounts of every mat in the game by scooping each and every set of 3, one by one. Ain't nobody got time for that nonsense.