Yup, STBXH is trying to pull wool over your eyes on that one. WH, the reason for that the head of household language is so you can get a greater tax refund by claiming EITC when filing your federal taxes.
I'd leave the language as it stands and your lawyer is absolutely correct in advising you to insert it in the agreement. Your L is truly looking out for your best interests...as your children's too. Listen to your L not some loopy guy in a raspberry beret!