remote start keeps grinding or shuts off

More info is needed - what year/make/model is your car?  What did you install in it?

You say 7v pk-pk would that be 0-7vdc or -3.5- +3.5vdc?

I would use a different tach reference - like a fuel injector.

The problem you are having is definately a sign that the tach signal isn't being read reliably.  Remote starts, since they are grounded to the battery, arn't capable of reading (-) voltages so your signal, to the remote start unit, becomes a (almost perfect) square wave with a 3.5vdc potential.  In theory 3.5vdc should be enough voltage to trigger the RPM input on the unit, but it obviously isn't.

When you say the wave is "not perfectly square" what does that mean?  Are there some quick pulses at the beginning or end?  That could also be why you are having issues - especially if the unit averages pulses.  Also, is the signal repeatable or does it have an abnormal signal back (like a crank or cam angle sensor would).

By going with a fuel injector you'll have a cleaner signal that will switch between 12vdc and ground and the remote start unit should have no problems reading that. 

The only other thing I can think to try is replacing the brain - maybe there is something damaged causing it to intermittenly screw up.

Almost any time a remote start system tries to crank the starter while the car is already running it is a tach related issue.  If the car starts and shuts off right away it could be a factory security issue or a tach issue.  However, since you said the car would start and stay on and recrank that is why, to me, it sounds like a tach issue.

Tach wires shouldn't be hard to find, just find a fuel injector and follow the wiring.  Tach wires may be hard to get to, but they shouldn't be hard to find.