"Got to" is the same as "gotta" it's just that native speakers rarely pronounce both words in "got to". So we write it as "gotta" to make the reader imaging how it's being said. Basically, there is no difference.
I think your lyrics phrase is fine. You have a lot of poetic licence when in this regard. However, you might consider "gotta put up that list of doing" if it fits with the music.
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