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i've run the battery down to dead a few times before I got the wiring redone. And unfortunatly the auto parts store's little machine says it checks out and won't replace it

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Slipknot -
I had a similar situation a few years ago with a battery from Autozone. It drained heavily a few times when it was new (I had a relay coil that was wired to be battery hot - OOPS!) and I kept recharging and using it. I fixed the relay issue but my battery trouble continued -- every few weeks, teh battery was dead. (I have a mini-starter so teh car always starts pretty easy even on low voltage).

Had the battery checked by the store a few times, it was always said to be OK. I had them check my alternator, I replaced some suspicious wiring connections, always had battery trouble every few weeks. Put new VR and alt on. Still had trouble. They pointed to teh car repeatedly because it surely couldn't be their battery, 'they hardly ever have them come back'.

Finally -
I charged teh battery fully, wiped all teh dust off of it to prevent old-wives-voltage leakage, and placed it on cardboard on my workbench. It went down below 12v within a week. Went to a different parts store chain, gave them that POS battery as a core, got a new one, and my problems went away.

Now, with all that said...
You've got 13+ volts at 1400+rpm. Even a bad battery is going to hold onto some of that and ramp down after you close the throttle, it's not going to plummet to 11.7 immediately. But the engine dies immediately. If the EZ-EFI is that sensitive to voltage fluctuation then it's a poorly-designed system (I suspect that's not the case, though). EFI designers know that they need to install voltage vs pulsewidth curves for the injectors to allow an engine to keep running for a while if the charging system fails (and also due to voltage fluctuation due to electical loads). I don't see this as your specific problem.

Can you set the idle speed up to 1400 with an adjuster screw? Get it running while holding the throttle, adjust hte screw up to that speed, then back it off slowly and see what happens.

It sounds to me like you have a fuel delivery problem. If it is due to voltage drop, then the injector PW is set so far on the lean side that the reduced voltage cuts the fuel too far. (which would mean 2 problems, really).

But yes, the easiest thing is to swap in a known-good battery for a 15-min experiment.