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I would first like to answer that yes, you can have Braintree (and any other payment integrations) on any number of forms. The only limit related to payments is how many you can accept per month, nothing else.
Having that said, the Fetching plans screen should work.
Also, thank you for letting us know that the form in question is the following one: https://form.jotform.com/60175154192149
I do only want to ask you if you could add the BrainTree integration back to the form and try to see if the same issue would be present for you. If it is still present, do let us know and we would check it out for you and raise it to our developers.