No attorney would answer a "should" question without knowing all the facts of a case in-depth. You appear to be correct, and the Bounds of Advocacy specifically forbid using child custody for either financial leverage or vindictiveness.
That said, you only have the options of capitulating (which does not seem reasonable; I presume you need the child support), or filing the motion to change paper custody to match the actual custody. It would be best if you had the "could not handle the children" notice in writing, but you can use what you have (your word, and the record of what has actually been done since then) against his version of the story.
Under Rivero (to review the case, see http://willicklawgroup.com/child-custody-and-visitation/), the Court will normally do a "one year lookback" to determine what custody has "actually been." It has been less than 6 months since the custody schedule changed, so it is difficult to know with certainty how the Court will look at the situation; it is possible, with the schedule you describe, that the court could find that you have been "de facto" primary custodian for half a year, and preserve that status; or, not. To answer your question, a full year would have been best, but it does not appear that you have the luxury of that much time.
If you do not have adequate legal counsel, you should get it -- these cases are highly fact-driven, and pretty complex, and few are able to competently go through them without assistance of a family law specialist.
Answered on Apr 03rd, 2015 at 12:54 PM