We’ve found some keys (a few sets, actually) one in Beitar and one in Yerushalayim.
We hung nearby notes and actually advertised in Beitar. We didn’t find the owners.
I want to know whether we can discard these keys. (My wife also is interested in just giving them to our younger kids as toys.)
What is our obligation, and what would the Rav recommend (perhaps lifnim mishuras hadin?
In this case I would give the keys to the kids to play with, and not worry about it further.
You have fulfilled your obligation of trying to find the owners, and there is nothing further for you to do.
The rationale for this ruling is a little complex; it is possible that the formal halachah here will be that the keys should be kept “until the coming of Eliyahu.” For keys, this is a very impractical, because experience shows that they ultimately get lost. The advise of Rav Moshe Feinstein, who writes that the item and its value and description should be written down in an “avedah notebook” is certain applicable.