Tom,
Your chassis batteries may not be bad yet. They are just being discharged and not being recharged until you start the engine, from the way that your description sounds. However your chassis batteries will be bad soon if you keep discharging them so much.
Your coach should have a way to charge the chassis batteries when you are plugged into shore power, but still isolate the two batteries from each other. SMC used an Echo Charger to accomplish this and Monaco used the "Bird System" to accomplish it. Since I have never worked on a 2009 Monterey, I can not say how your coach is set-up, but what ever system it is equipped with, it is not working if the inverter is charging the house batteries.
The "Bird System", if equipped, is a system that ties the two battery banks together when either one of the battery banks is being charged. That is a simplified explanation because there are voltage thresholds that need to be meet and time delays, but that is basically how it works. The "Bird System is not a battery charging system, it is a battery isolation system that allows both battery banks to be charged if either bank is being charged.
If your coach does not have this type of system, you need one because the engine computer will ruin you chassis batteries by repeatedly discharging them. There are similar systems that are used in the marine industry that are probably better, but I will let the boat people on this forum handle that. If the system is there it needs to be repaired.
As for your question about the bottom of the slide, metal reinforcements are done, however they need to be done right by someone who is knowledgeable about the slide or you can cause slide and/or seal damage.
I know a lot of people like Camping World for service work, but someplace like Beaver Coach Sales is so much more knowledgeable about your Beaver coach that you will probably end up with better service at a lower price.
Gerald