This is a very interesting question. A simple google search found this flyer for the annual Philmont fly fishing invitational. https://www.philmontscoutranch.org/wp-content/uploads/2020/07/Fly-Fishing-Invitational.pdf. The folks that organize it would be good source of information.
You could try posting the question on other Philmont groups. For example, this one
[email protected]
If your itinerary has some layover days, you should have time to fish. If some of your backcountry camps are close to streams, it seems like you could fish in the late afternoon or evening if your crew gets at early start in the AM.
I myself would have appreciated supplementing the Philmont fare with some freshly caught fish.