I think Pete and Rikki have got it right about the exercise, but you need to hear and play without translating trough several layers in order to play fluently. If you hear something (like happy birthday), then have to analyse it by saying to yourself OK all is fine until I get to those big intervals, what are they? Oh yes, when I play in my key of C the first jump is an octave above the G (start note) then the next big jump is an F (the forth). Ok they are playing in Concert E that means that (since I'm playing a tenor in Bb) I'm in F so when I get to that part I'm playing a C octave jump and then the forth which is Bb. Yea everybody does that! Right!
You hear it in your head and you play it. Analysis is great for developing understanding and OK after the fact or if you need to figure something out well ahead of time. A pro just plays it, no analysis, no picture of notes on a page, no transposition as a mechanical intervention. You hear it, you play it. If you try to analyse everything on the fly you’re dead in your tracks. If you are trying to make this somehow a fast process when forever translating it’s just as awkward as trying to translate any other language through a dictionary, hand held translator, or even another person who translates. There is a time delay and something always gets lost in the translation. Why not be fluent in the language in the first place? When you work with your ear and your instrument to make them one, then you can play what you hear. Don’t know about any of you, but I think I can sing happy birthday to you in all twelve without thinking about transposition or structure. Can be just as easy to play IF you train yourself instead of trying to forever translate through an intermediary.