Yep, welcome to the worship band world. There is no excuse for Eb or Bb!!! LOL I never use capos on electric, kinda defeats the point, but those kinds of changes change the whole feel of a song. Just shift it up or down a 1/2 and live with it, please.
We used to play Today is the Day, and I did learn the solo pretty much note for note, except for that little run before the end melodic part, I usually just threw a scalar run in there to connect the 2 last "sections" of the solo. It's not that hard if you break it down into it's logical sections and learn them separately. Unfortunately our church has now deemed it as "old" and we don't play it any more, but IIRC, the beginning section is melodic and doesn't move too fast, it has some greasy bends and slides, good stuff. The next section has some speedier repetitious phrases (again, it's been a long time), make each of those an exercise, and use hammer ons and pull offs, he isn't picking every note, not worth trying unless your last name is Morse or di Meola. My memory is a little fuzzy but I think it goes into a run in 5ths or something after that, you can just use a 3 note per string scalar run in A (again IIRC), then the ending is a slower melodic passage on the 1st and 2nd strings around the 12th - 14th frets.
Oh yeah, the biggest thing to pull stuff like this off, RELAX, if tense up it won't happen. That's why it's best to work up the little sections until they are automatic and you don't have to think about them, they just happen. I'll help any I can if you want to PM me. I think Lincoln even shows most of the parts on a utube video.
Also, as has been noted, it's worship, it's hard to refuse to play something, but you can tell them if you are uncomfortable. Also don't feel like you have to hold yourself to a note for note solo, that's not the point, play blues licks in F# if that works better for you (assuming you are still playing the song in A).
FWIW, I miss playing TITD, pretty much the only worship song with a solo that's ever been in our setlists. About all I do now days is play little sprinkle parts between words.
About pitch shifting, software is great and I use it even for learning "normal" stuff in Eb so I don't have to retune, but for the worship stuff, if you guys are using planning center or something they should be putting the songs in there already pitch shifted, I have had to shift songs in the past, but there are so many resources out there for the worship leaders now days there's not excuse for them not providing them in the right key. Well, it may sound like the chipmunks or Lurch, but still.