Weird tempo map question?

[kmcnair1]

I have imported a a larry carlton tune into Pro tools 12.6 and analyzed and done a pretty detailed tempo map and recorded all my parts to the feel of that tune. it feels great. Just like the tune . The problem is I'm a really bad keyboard player and need to slow the song down to be able to record the Piano parts. But when I slow it down, say by a 1/3, the tunes starts speeding up and slowing down drastically when its following the tempo map... so bad so that you can't even play over it. is there any way to lock the tempo so I can slow it down and then revert the tempo back to the tempo map or print the tempo map. this is very hard to explain. Any Ideas on how to smooth out the tempo so i can slow it down and record the piano and then speed it back up later and it feel the same?