Long before Lee came out with their sizer method I sized the same way in home made dies. It works quite well for those on a bugit.
I recommend pushing gas checked bullets through nose first, so the check gets clamped on tight by the pusher pressure. I've never seen a Lee sizer so don't have any idea how close the pusher punch is to sizer diameter, but for pushing gas checked bullets nose first the pusher diameter should be no less than .020 smaller than the sizer, and preferably about .010 smaller, so the checks are held square and can't cup out or deform. Size your gc bullets without lube, so the check is locked on and lube can be smeared on easily with the fingers. Lubricate them with LBT Blue soft and they will stand more velocity and pressure without leading than with any other lube you can buy. -- You'll quickly learn about how much lube to put on, doing it with the index finger of your master hand, then holding the finger out of the way while sizing. Wipe excess lube off the base and nose ahead of the sized area with a paper towel or rag, and your bullets will look about as good as if sized and lubed on a lubrisizer. This covers all the requirement you listed. It is the easiest and fastest way to hand lubricant, and less messy in my opinion. Not as easy as using a lubrisizer, nor as fast, but very efficient and effective when the very low cost is considered. When order lube, get solid sticks so you'll get a bit more lube. We advertise 10 sticks as the minimum order but order two or more at $3.50 each and we'll go for it.
You are correct when you say liquid alox is a mess and not a very effective lubricant. I won't expound farther than that as I might sound vindictive or something!
