I sent her a long text explain all that we'd been up to this weekend.
Then, earlier this morning I texted her to let her know the baby was doing fine and sleeping well. She responded that she was "glad to see I was doing all of the things she wanted us to do".
My suggestion is that whenever she makes these snide remarks, you cut back on how much you inform her of what all you and the baby does. After all, you are his father....not some stranger who has taken him. If you need her for anything, you'd contact her, but to report to her throughout the time you spend with son is not fair. I'm sure she would resent very much if you tried to turn the tables on her.
Since she made these comments, you might want to hold back initiating any futher contact until time for the child to be with her. Maybe, just maybe after a few times of seeing that she can't talk to you like that without expecting some sort of consequences, she might stop with the sarcastic statements.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!