If someone has anaphylaxis from a certain drug. if they melted it in their mouth could it trigger anaphylaxis without swallowing?

Yes: They will react in the same way regardless of how it is taken.

Yes: Anaphylaxis can be elicited even by very minimal amount of offender. As an example, we learned a long time ago that even a skin test might elicit anaphylaxis and could not be used as screening tool for penicillin hypersensitivity.

Anaphylaxis: it can trigger anaphylaxis even without swallowing.