I hope there's a successful player(s) here with perfect pitch that could give advice.
It takes a long time for me to get used to trumpets in new keys, because I rely heavily on the absolute pitch, rather than relative pitch. Since I started with a B-flat trumpet, playing on that instrument is second nature. I learned the C trumpet a few years ago. It was a real hassle at first. I remember being assigned a Charlier exercise and it was a nightmare to learn. Today I can play with the C trumpet almost as effectively as a B-flat trumpet, whether playing a B-flat or C trumpet part. However, I have difficulty with rapid passages on my C trumpet, especially atonal works, unless I examine and practice them.
A couple months ago I picked up the E-flat trumpet for the purpose of playing the Haydn trumpet concerto with it. It took me weeks to even get the point that I could hear a given note in my head and actually play the correct fingering consistently! (I performed the Haydn just two days ago, and it actually went fine; I only had two or three fingering mistakes; of course there were other kinds of mistakes, too.... nonetheless I can play it better on the E-flat than on my B-flat)
It's weird. I can apply relative pitch when I'm transposing, but not when I'm playing a transposing INSTRUMENT (unless it's a B-flat, B-flat piccolo, or C trumpet). Any ideas to improve it? Perhaps I should apply solfege?