The transfer switch is what is in your earlier images that has a label stating Model ATS 301 (ATS = Automatic Transfer Switch). Its purpose is to switch between the shore power and the generator power automatically. Thanks. Now that I read that, I knew that. Just too many new terms bouncing around in this old noggin.
It wouldn't be bad though if your RV has power to 120v circuits such as the Microwave or TV when on Shore Power as well as on Generator Power (without shore power). Can I rephrase that and say since all the 120v things (refrigerator, microwave, outlets, etc.) work, both shore power and generator, the ATS is probably not the problem. Don't have a TV.
If you believe the battery is bad, most battery resellers can test it for you for free. In my experience though, the testing often indicates a battery that may still be good but generally not as good as it was when purchased, therefore a battery that tests out as marginal often is said to be good if under warranty and bad if not under warranty. In reality and from your reports, I suspect it is still good but by itself not enough capacity to run your RV for very long.
If you intend to use your RV often on battery power only (no shore power) then it becomes more important to consider your options with a new battery, perhaps a better battery like a Lithium LIFEPO4 and\or multiple batteries. On the other hand, if you rarely camp (RV) on battery power only then perhaps the battery you have is just fine especially if you limit your 12 volt usage. I try not to depend on battery, even to the point of not using the lights, just a flashlight. But a better battery has been on my thoughts. What battery can I use and not make any other changes? Doesn't a LIFEPO4 require some hardware changes?
Keep in mind that a furnace blower (if you are camping where the furnace is running often) will likely be what uses the most of your battery in overnight usage. ~CA I have run the heater once or twice for a few minutes just to check it works. It does.