Well, not a competitor's fare, but an Orbitz (or Travelocity, etc.) fare for an AA flight, since this was in the OP as well. (Yes, I know that this has been discussed ad naseum. )

I, and others, have done this. Doesn't always work, but it can:

Find a fare on Orbitz, etc. on AA. Note the fare bases for the segments.

Call AA web services (for me they handle this better than PLT line, at least). See, first, if they find the same fare. Likely not, or you wouldn't be calling.

Tell them the fare bases that you found.

Often, when presented with these details, they will be able to find this fare for you.

Again, it has worked, and not worked, at times. It's always worth the try.

Cheers.