Hi!
Just catching up to the postings.

The part about the battery showing 12.4 volts without the engine running is not very good.
It helps explain the 11.8 reading with the engine running. I bet the fuel management system did not like it one bit as this can contribute to poor performance.
The battery needs a good long trickle charge to put your readings in order with a good alternator setup that I see you are looking for in your garage.

12.4 means to me that battery is run down or is getting long in the tooth, per say, age wise. Not a good habit to operate a car under this time of year.
Give it a good charge and do not expect the alternator's system to bring it all the way back up to a good longer life if you don't!

FYI,
Car batteries are not deep cycle built. Meaning they are not to stay in a "low charge state" over long periods of time.
They do not have the extra depth, below the plates, for the sulphating debri to collect that eventually shorts the cells more quickly on regular car batteries. Deep cycle batteries are made more robust to take the temperature swings of high discharge or repetitive charging cycles too.

In most cases, a deep cycle battery is a waste of money unless you depend on auxiliary equipment to operate during off cycles of power plant support systems.
Driving a car regularly, is a must, for so many reasons to numerous to name above and beyond batteries but batteries are the first to suffer, very quietly!

GAS ranks number two as it just looses its "fizzle" over a longer time interval.
(-:)
Phil