Which glue will work best to repair plastic dome lid locking part?

The best advice I've seen is here. If you click "Where can I get this part?" , it suggests taking the broken pieces to a retailer and seeing if they can fix it. That is, as @salmonjapan mentioned, if the are comes into contact with food at all. If not, there are many possibilities to fix it yourself.

The last review on this site is from September 2014. It's been out of stock for awhile, so I wouldn't advise chasing after a new part.

As for figuring out what kind of plastic it's made of, this website has a fairly long article explaining the different types. Be aware that there is a burn test involved. Here's the same information, but in a concise chart. Without burning anything, the article here suggests perhaps polypropylene.

If nothing else is promising, you might actually try plastic welding. You don't need a saughtering iron, you can heat up a knife on the stove. How hot the other end of it gets depends what the knife is made of, silver being really conductive and also valuable so I'd avoid using a silver knife, just in case.

In looking into the plastic welding more, I found liquid plastic welding products. This video explains how it was first developed by a dentist, though they're not specifically advertising using it on food products. It is not FDA-approved as food safe. Still might be worth a look, particularly if the part never comes into contact with food. Here's a comparison between two similar products.

Good luck!