You're not overthinking this at all, it's a great question!
I do not recommend thinking in terms of the intervals that you gave; that ruins the sense of large context and the overall flow in the music, and your performance will often come across as stilted and disconnected.
Instead, try to find larger patterns. The good news is that you already did: this chart is just a succession of three ii--V--I patterns, each one a whole step below the previous! This means that you have the following ii--V--I patterns for each transposition of the piece:
- In D: D, C, Bb
- In Eb: Eb, Db, Cb (=B)
- In A: A, G, F
I recommend separating the hands first; make sure you can play the harmonies in the LH before adding in the melody in the RH (assuming that's how you have it orchestrated).
Then you're just transposing the melodies themselves. I recommend thinking in terms of scale degrees: the opening three measures is 5--4--b2--n2--3. You might think this is the same as your intervals from earlier, but it's different because this way shows how they all connect in a given key; the intervals from earlier could have changed keys every beat.
Of course, you'll use the same process for the final phrase.
You mentioned you'll probably eventually transpose it to all twelve keys---you're probably right! So get that ii--V--I progression in the LH fluent in all twelve major keys, and then get that opening 5--4--b2--n2--3 melodic line fluent in all twelve major keys, too, and before you know it you can play Tune Up in any key you want!