I'm a little ashamed to say it took me this long to figure out something so simple.
When I make doors I only have one shaper to handle the coping of rails, (vs having two setup so you I have a proper left and right copes). I've always taken a ~9" wide board and put it in the cope clamp and ran the profile on it, then rip a chunk off for a backer stick while coping any sticking that has a profile. You don't need anything for flat/shaker style doors.... The way I cut my sticking I cut it long an 1/8", and the coper takes off a 1/16" on each end, so my backer sticks always get shorter, and eventually they end up too short to use.
It finally dawned on me to take the coper head out of that shaper, and put it in the shaper that I use for machining all of the sticking. Its already at whatever height, and its a perfect match for the sticking that has been already run, plus I can run much longer pieces than the 9" that fits in my coping sled. Really at the beginning of each job when making doors I could grab one of the ugliest rips and send it through and have plenty to spare. Unless I was doing something idiotic like 200 doors at a crack, an 8' rip should be adequate for some time. Or, just grab a chunk of whatever length and toss it through the shaper.
Damn the dumb stuff gets by once in a while.
I think this is a case of it worked well enough, that I didn't really put any thought into it until I was mindlessly doing something else here and it popped into my head.
Also, I know what some of you will say, just do the stick cut after the rails have been cut to length and coped, then all I need is a square backer to prevent tear out. That is do-able, and I have done it that way at other places I have worked at, but it also introduces a lot of chances for things getting bunged up in the sticking process, especially when doing short(er) pieces.
Where's JR Rutter?