In the USA, you normally only need to send 3 letters of recommendation:
- One of them should be your dissertation supervisor (if diss < 5 years old)
- One should be a current supervisor (such as your new postdoc advisor)
- One should be from a committee member or someone else who knows your work well
As long as you have three, I think you're fine in not including a letter from the postdoc advisor that you're having problems with.
tl;dr: Letters of recommendation are used to indicate that some (at least three) faculty think highly of you, not necessarily that all do.