[oil pan 4]

The term "full charge" is a little tricky.
That means the voltage the battery retains after it has been sitting for 4-8 hours (depending on the mfgr.) with no loading or charging and at a temperature of 70-72'F.

"what is the threshold at which the alt considers the battery too low"
I can't say for sure how your system works, but it likely adhears to general starting battery charging guide lines. If I was going to make an alternator system that cycled on and off to save fuel I would set the on voltage for any time voltage dipped below 12 to 12.2 for more than a minute and off after maintaining 14.5 for about 10 minutes. Also have the alt run for at least several minutes on every start up.
Also your deep cycle battery should maintain higher voltage even as the battery power is consumed. For example lets say 12.4 volts on a starter battery could mean its 90% charged while 12.4 on a deep cycle could mean its 75% charged.
Unless your battery gets real warm it should never develope more than 12.8 volts on its own, after its had time to sit for a few hours.

I only get to connect my fancy over priced charger to starting batteries, I cant be sure what the fine details of deep cycle charging look like. I do know starting batteries will charge and discharge fast and hard enough to burn them selves up if the power or the load is there.
I am thinking that the deep cycle could trick the charging system on your car into believing the battery is charged when its not. Becaues I have herd that deep cycle batteries charge and discharge slower than starting batteries at the same voltage and on the same loads.
We know that Deep cycle charges run at higher voltage than a normal car alternator system. I believe the normal charging voltage on a starter batt charger (plug in unit) is the same as a deep cycle charger (a little over 15 volts), but the deep cycle chargers have an added feature. An "equalizing cycle", where they push the battery voltage up to 18 volts and cook the cells a little so they all get a chance to take on an equal amount of capacity. I would always use the equalizing cycle after other people ran the batteries in my tugs I use to take care of way down to the point where they would just crawl along.