You can cut out the shape from the plastic top of a spray paint can to fit the indentation and glue it in place with contact cement. It will wear a long time---especially if you put a patch over it. All you need is a steady hand and an exacto knife.
I rebuilt a HR mouthpiece with waterweld after it was recommended here. I've also replaced the bite plate on a brass piece. It's very easy to work with and sticks well. I keep the repairs covered with a patch to protect them from my "beaver" teeth.
Waterweld wears very well if you put a patch over. It's certified by the US Food and Drugs Administration as safe for use in vessels containing drinking water, so you can be sure it's safe in your mouth. I wouldn't put normal JBWeld in my mouth.
Not sure I would want epoxy in my mouth. It's OK further down in the mouthpiece. Mojobari uses ordinary JB Weld for mouthpiece repair. Including chip repair on the tip, come to think of it. But it's a bit powdery and I think teeth would wear through it in no time.
I've used old film canisters in the past. They aren't so readily available these days so something like John's suggestion might be good. But you have to be careful with plastics which can absorb or leech chemicals. Maybe try those black ice cream boxes?
I've known people to cut down an old mouthpiece and make an ebonite bite plate. It's a lot more work but you might prefer the result.
I guess you can carve them in ebonite from old mouthpieces or clarinet bells and have a "vintage" result.
Pillinger would use the same compounds ("resins") he uses for his mouthpieces, but I don't think it would be a cheap solution for just a couple of pieces.
Waterweld is probably the easiest solution. For myself I would use Milliput and a patch.
Manny - A couple of ideas.
1. you could make an ebonite plate from an old mouthpiece or from ebonite sheet .
2. you could use dental epoxy.
3. You could get a denture maker/repairer to do the job for you.
4. You could try one of those acrylic nail painting parlours.