@Georgia - Either the tax advocate does not understand the situation or the situation is not as you stated.
Because you are not the poster that started this thread and your circumstances are different than the original question, I suggest that you start your own question. I am closing this thread since it has wandered off-topic from the original question.
The only time the tie-breaking rules (whose income was higher) apply is when two tax payers both meet the requirements to claim a Qualifying Child and cannot decide between themselves who will claim.
To meet the requirements to claim a Qualifying Child the residency test says: "3.The child must have lived with you for more than half of the year.2"
[footnote 2 says: "2-There are exceptions for temporary absences, children who were born or died during the year, children of divorced or separated parents (or parents who live apart), and kidnapped children."
IRS Pub 17 <a rel="nofollow" target="_blank" href="
https://www.irs.gov/pub/irs-pdf/p17.pdf">https://www.irs.gov/pub/irs-pdf/p17.pdf</a> page 26-27
You say the child has lived with you for 3 years and not with the mother at all, so the mother cannot meet the residency test and CANNOT claim the child as a Qualifying Child. (The footnote 2 exceptions do not apply since this is not temporary absence and you are not one of the divorced or separated parents).
The other way for a parent to claim a child is as a Qualifying relative which does not require that the child live with the parent at all, but to do that, the parent must 1) provide more than half of the child's total support for the year, AND 2, more important, the child CANNOT be the Qualifying Child of any other taxpayer. Since the child can be YOUR Qualifying Child, the parent fails both tests 1 & 2 the parent cannot claim the child at all if the facts are as you state..
That is basic tax law and anybody claiming to be a tax advocate should be well aware of the basic dependent requirements.
I say again, that there seems to be a mis-understanding or disconnect between you and whoever you have been talking to.
**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**