Hi Dan,
That's good info thanks!
Yes you might be right. Do check the Escape wheel teeth, pinion and pivots as well as the rest of the escapement action.
I'll make mention of a couple of other possible things that I check just in case.
For now I'm going to assume the same as Dan about you meaning one revolution of the center wheel per 9724 BPH and not the escape wheel. So rounding off to 9720 would give 162bpm. So with an error of 4 BPH and using some quick math <warning, alert, caution! Don't trust my math!!!> would put the rate error somewhere in the neighborhood of 4+minutes for seven day run. So yes your 3 day rate variation ranging from 2 to 4 BPH is dead on. So,,, if that's correct then we've actually determined two things here; your clock is getting close and I can still count!
Now for the clock only running half of it's intended 8 day run, well that's a problem that needs to be addressed for sure. Could definitely be a set or incorrect mainspring as you suggest or something like a slightly bent tooth somewhere in the train. A slightly bent tooth can be difficult to detect and often won't reveal itself by manually moving the train. Also it takes some force for the time train to activate the strike so there could be an issue there too. Something that I do when this sort of thing happens, and it does, is at the point when the clock stops I place a small mark with a marker pen or whatever at the point where all teeth mesh with one another in the time train. I then deactivate the strike by either running the movement without the strike train installed or by raising the lifting lever so it's out of action. Then fully wind and let it run until it stops again. If any of the marks line up perfectly then that tooth or teeth are suspect. If the marks are close but maybe not exactly in line then it's possible there's a slightly bent pivot on the meshing wheel or pinion arbors. Finally if the clock runs through to the end and the mainspring looks good then check to see if someone replaced the original spring with one the wrong size. This is common especially with vintage and antique watches. If you figure out why it's stopping prematurely and fix that problem you might also find that the clock rate levels off enough to time it within the 2 minutes or less which should be acceptable by most customers for this type of clock.
That's pretty much the way I go about it but I'm sure others have different techniques they use for solving these problems. The order that I do things can always be changed around to fit your own needs and likes.
I'm mostly a watch guy going all the way back to the 1960s. In fact Bob and I met in the early 1970s when we worked together at a Hamilton repair center in Woodland Hills Calif. Whew! So you will find other more qualified clockmakers up here. Many of these same techniques apply to both though so I hope it helps Dan.
Please keep us posted on how you make out. Best, Dean