Let’s start over. You state a fully charged battery 13.6 VDC will spin the engine over fine and start the mower. If you let it sit for a week your battery voltage only reads 12.3 VDC and turn the starter very slowly, won’t start the engine. If this is correct, I would be doing the following, in this order.
1. Fully charge the battery, install your VOM across the battery terminals and start the engine. As it is cranking read battery voltage.
2. Once the engine starts, at full throttle, take a second battery voltage reading.
3. Let the unit sit for a week, connect your VOM across the battery terminals and read battery voltage.
4. Now try to start engine and take a reading while cranking. If it starts repeat step #2. If it doesn’t start, try pull starting and repeat step #2.
Let us know each of your readings. From them we should be able to tell you if you have a bad battery, bad stator or a voltage drain in the system.