I assumed already that the task to have one button for all weapon types is a bit more complicated for you but never the less the best solution.
But like many say it here already (incl. me) to get both attacks, melee and ranged would be also a solution and certainly not hard to do for you.
And another option, if you dont like to give a character both attacks (personally I see not a big balance issue) is if you would give a third option that only starts the bird attack without any weapon use and requirements.