It's not clear whether you've yet tried using a double-legged kickstand, i.e. the ESGE/Pletscher one. That's the natural solution for an application like this. That can work with or without a plate, but if so far you've only used kickstands that require leaning the bike, you should probably just get that first.
Chainstay damage tends to either be the result of the kickstand being way overtightened, or of it coming loose due to being undertightened/underlubricated to begin with. It can also result from chainstays that are too thin to have a kickstand clamped on, but with a bike like this that's unlikely to be the case.
Yes you can have a kickstand plate brazed or TIG-welded on. Brazing is the superior choice here, since welding has no advantage and the disadvantage that the effect on rear end alignment and need for subsequent correction will likely be more severe. Really either would be fine though as long as it's done in the hands of someone that knows what they're looking at. Adding a plate is the kind of joint that requires a good feel for when the tube part is reaching active temperature, since tubes soak up heat from the torch while the edge of the plate will get there much faster. It's not rocket science but it's also not level one. Don't be optimistic that it can be done without the need to check and correct the rear triangle alignment, and also likely removing the stoker BB.
Edit: On second thought, adding a double-legged kickstand to the chainstays probably doesn't actually make any sense here, since what you really want to do is load up the child seats with the bike standing up and well-supported, then have the captain get on and kick the kickstand back. Once the kids are on, the front of the bike will be floating and the ground contact spots on the rear end will be relatively close, which is not very stable. You should have a framebuilder put a plate behind the captain BB on the boob tube. That does everything you want. That way the load is going into the rear wheel where it belongs and the kickstand legs are stabilizing the bike but with little load themselves. You'll need to get the chain clearance worked out first, but the ESGE kickstands are made to be good at that.
