Capacity testing, part 1...
So far it looks like
all the smaller 7.2 Ah batteries are probably duds.
I tested the "best" one first (the one with the starting/resting voltage of 13.1), using a 40 watt, 120 volt incandescant light on an inverter. It pulls 3.7A from the battery at the start (when batt voltage is high), but it goes as high as 4.15A as voltage falls close to 11.0. The inverter shuts off automatically when the battery gets below 11.0v. So let's call it a 4 amp load.
Note that different inverters vary on their shut off points. And some capacity tests go as low as 10.5 volts to evaluate a 12v battery - eg.
How to Measure Capacity – Battery University
The "best" 7.2 Ah batt could only deliver
4 amps for 7 minutes before the inverter automatically shut off when the battery slid down to 11.0 volts.
---
The 12 Amp hour batteries look more promising. I've been pulling 4 amps from one of the pair for 45 minutes now, and voltage is at 12.06 and falling reasonably slowly.
EDIT: pulled
4 amps for 53 minutes before the inverter shut things down @ 11.0v. Obviously voltage falls much faster at the end of the discharge - but you probably already knew that!
---
It'll be interesting to see how the mower treats the batteries: what amperage the motor pulls under normal conditions, and at what voltage it shuts things down to protect the batteries (assuming it has an automatic shut off).