Sir, my name is ***** ***** I am going to assist you with this. I am a certified Generac technician, and I have actually ran into this problem before.
The first time I ran into it, I fought it for 6 months.
I just couldn't justify spending $100+ on a Generac charger.
The problem you are experiencing is not due to the size of the charger, it is due to the type of charger. The 1A unit should be sufficient to keep this battery charged - I use a 1.5A unit.
First lets back up a bit and go over some terminology.
What we are talking about here is not actually a battery charger - it is technically a battery maintainer. It will not charge a dead battery like an actual charger will. All it can do is to provide a small "topping off" of a slightly low battery, and maintain a full charge (obviously provided there is no load bigger than the rating of the maintainer).
The problem with most maintainers is that they are microprocessor controlled. They can "read" the state and status of the battery. If the battery is too dead, the maintainer will try to "shock" the battery to start the charge. But the problem is that the strength of the shock required is greater than the capacity of the maintainer, so the maintainer will lock itself out, until it is reset.
The problem with a microprocessor maintainer is that whenever the starter kicks in, the maintainer sees this as a major draw on the battery (which it is), and automatically tries to shock the battery. The starter is probably drawing 80A - 100A, and it overloads the maintainer, so the maintainer locks itself out, until it is reset.
The solution is to find an analog maintainer, instead of a microprocessor maintainer. An analog unit simply puts out its rated amperage, regardless of the load on the battery.
The Generac maintainer is a microprocessor maintainer, but it was designed and built to Generacs specifications. It has a timer built in, that when the starter kicks in, the maintainer does not automatically start right away. It waits a specified amount of time before activating, to allow time for the starter to stop. It also has a charge timer. Once it starts charging, it only charges for 18 hours, and then shuts down. It sits in a ready state until either a load is placed on the battery, or due to parasitic draw, the voltage drops to a specific level. Then it will kick in again. The Generac maintainer puts out 2.5A simply because it does NOT run all the time, like an analog maintainer does.
The maintainer I use is the Schumacher SE-1-12S. I am sure there are other analog maintainers out there, but I quit looking after I found the Schumacher. I have installed them in 6 or 8 units over the years, and have yet to have trouble with one.
By the way, there is also a Schumacher SE-1-12S-CA. It is not the same charger - the CA version is a microprocessor unit.
Of course, you can use a microprocessor maintainer, but the drawback is that it would have to be reset (unplug it for 5 seconds, and plug it back in) every week, after the exercise cycle.