It's awesome that you're looking for an economical way to do this and thinking outside the box.
In this case, I would not weld the two halves together. It's a flexible joint for a reason. (You don't want your cylinder head to become the weakest part of the exhaust system!)
From the pictures, you are beyond a standard fix of the flexible joint, because half of it is missing! I recall that the plate is part of the catalytic converter? So your best bet may be to find a muffler place who will cut off the old flexible joint and fabricate a new one, which may or may not be possible for them to do, depending on where you live - emissions testing requirements vary between states.
Also, check RockAuto.com for catalytic converters, because maybe by the time you have paid for the labor, you could have replaced the catalytic converter. However, my understanding is that some of the aftermarket cats are not legal in some states - CA, NY, CT, I think. Someone else chime in?
When you replace the flexible joint, you will want to replace the donut gasket AND the springs and bolts. If the threads are shot on the plate the bolts screw into, drilling them out comes to mind, and may be what some shops do to save time??? Ugh. I don't think drilling out the plates is a good idea. I suspect it might cause an early failure of a different kind, for example, the bolt wearing against the plate as it is constantly being vibrated, wearing thin, and breaking. Thus, I used the method described below to avoid having to re-tap the threads.
You can buy kits with the bolts and springs. If you Google this site (add "site:insightcentral.net" to your search terms on Google) you should be able to find part numbers from people who have done it.
Somebody posted just last week about a suitable, common, very inexpensive replacement for the donut gasket which is quite inexpensive.
Before removing those bolts, I sprayed them every few hours with PB Blaster for three days before trying to remove them. If you crank on them right away you WILL break them off. Instead, alternate slightly tightening, slightly loosening, using the pressure to gently break free the corrosion. Work it very slowly. Eventually it will start loosening. Keep tightening, loosening a little more than you tighten, etc etc. We are talking about fractions of a millimeter - you are LISTENING for movement, not actually seeing it at first. It will take a while to get them going.
Check the rules for your state's emissions inspection to make sure your muffler repairs is going to pass the inspection if it is anything less than replacement with Honda parts.