Tune the blades to the bed knife and then backlap them. I'm sure there are several ways to do it, but this is what the mfg recommends on most brands and is reliable.
The golf course compound is 120-220 SiC in a grass-safe water-soluble vehicle, meant to be used in just this fashion on gang reel mowers. If you have the facilities you can chuck the shaft in a drill press or clamp it and chuck it in a larger hand drill (be careful!). It will sling a bunch of grit but will get the job done faster. When the edges all feel "sharp" or at least can feel a burr, you're done. Test it by slow/manual cutting newspaper in small strips along the entire edge of each blade.
The cleaner the mating surfaces touch before sharpening, the faster it will go, and once sharp the lighter the contact can be set - easier to push. I usually start by setting the contact as light as possible and seeing which blades/sections of blades don't make contact - can't cut or even catch the newspaper. Next I beat those areas from the opposite side of the reel with a block of wood and a mallet - firm smacks, nothing crazy, test contact every time. Start by referencing the reel to bedknife using the blade(s) with the hardest/first contact. Once all the blades are 80% minimum good to the bed knife I go ahead, reverse the pawls, and lap the blades with compound.
Don't bother going to a super high grit, the steel is relatively soft to begin with, and the SIC breaks down rapidly when lapping anyway. It will hold the edge a long time, especially if you watch out for small branches. I used to mow the yard at one of my houses with a reel mower, was about 1.25 acres (IIRC). Properly tuned they cut great, but I have yet to own or sharpen one that didn't need tuning from the factory before sharpening.
