Well, there's solid intelligence that nationally there is a movement towards 'management' of parents advocating for federal protections. That said we can surmise all day long. What it will come down to is (1) getting back to procedure (2) gathering evidence along the way to document as if this will eventually reach a threshold triggering intervention through a complaint for safeguard violation, or subsequent SEA LEA complaint process for substance like no related aids or services in general education setting to access curriculum.
I don't know OP's facts but I would prefer framing this within hypothetical supposition because this conversation has no privilege from discovery.
Hypothetical #1 no history of anaphylaxis shown
District may cite a lack of documenting disability sufficiently to grant eligibility. Although LTFA qualifies the parent would need something that documents it sufficiently to school that child has LTFA. The nurse might be playing doctor (and, badly) by trying to infer whether epinephrine was administered, and when as a diagnostic determining eliciting dose or threshold, severity of past reactions as a reliable predictor of future reactions, and more worrisome -- possibly what symptoms out of the constellation of symptoms mean anaphylactic reaction in this student in order to receive treatment.
Is nurse crossing the line into practicing medicine in a field she has no expert knowledge in on a person who is not her patient?
Hypothetical #2 history of anaphylaxis but not treated with epinephrine or EAP
This would be a tough one to get through to them, but can be overcome with medical attestation and other written authoritative submissions to include as parent provided sources in the 'variety of sources' for them to 'consider', consider unfortunately lacking a concrete definition.
So, push - politely, but firmly. Insist on a copy of the district's handout to parents on the 504 process. If you don't understand something they say SAY SO EXPLICITLY that you don't understand your rights, you don't know how it pertains to 504 procedure and process, and any answer needs to be in writing. But you unfortunately have to reload this one to get it course-corrected. I bet as soon as you ask for anything like a handbook, what your rights are, where this all is in terms of procedure, team meetings, etc., their ears will prick right up.
Remember this, always: If it didn't happen in writing it never happened. This means even if you have to make a note to yourself on your phone to email back to yourself - do so. Create your own paper trail if they refuse.
By the way, I am not a lawyer and cannot give legal advice. I do have personal opinions as a parent of a child with LTFA and developmental disabilities. I have acted as my childrens' advocate through 504 and the IDEA protections.