The MSD uses both the stator and the battery, the zeel runs only off of the battery, so as yamanube said it cares nothing about the stator and nothing about the rectifier, they could be completely removed and the zeel would work fine.
The Zeel does need the trigger to be good, it is mounted in the cover outside of the flywheel (so separate from the stator).
I've had no noticeable change in performance with a nearly dead battery, and the specs on the zeel say it only needs 7 volts. Maybe there is some reason why it affects some setups and not others, I dunno. You're going to need to straighten out the battery thing either way so you might as well and see if it helps, but I'm guessing you have something else going on.
If it really is 15 seconds of strong cranking....that doesn't sound like all that bad of a battery to me. That's a hell of a lot of cranking, if you have any reason to be cranking for 15 seconds you have something very wrong with your setup.