One cog off.
Usually a engine will not even run one tooth off and especially come up to speed and carry a load and especially being a generator engine. Make sure you get the cogs right before buttoning it up.
The reason I asked about electric start is some of them emergency gens, especially 10 hp and up are really hard to rope crank in cold weather. (and requires a good battery, sometimes a small lawn tractor battery in cold weather will not crank them and being off grid to keep a battery charged may require jump starting from a vec battery.
You might want to consider taking the key switch to a locksmith and get a key made so as to have easier electric start cold weather cranking since you are off the grid.
UP/down flywheel key won't change timing. I thought you were indicating the key was partly sheared sideways which happens sometimes.
Strange that it is one cog off from the FACTORY.
You can review the operation of the ACR on-line. ACR when correct releases some compression for easier starting and not hanging on compression stroke like you describe. Automatic ACR is critical of correct valve adjustment, especially on a OHV engine. The ACR is usually built into the cam and slightly opens USUALLY the intake valve JUST BEFORE the piston is at TDC. (but can be on the exhaust valve)
Do you know if your engine is a Honda, Briggs or what make????
If you see any engine model, code, etc, post them. We can then get give you more logical precise repair/troubleshooting info instead of just ball park help. The gen model you listed did not help me to get engine info.