Just so I understand, you are looking to move from Grobb to Whitemane, right?
Wherever Grobb has Free Character Migrations right now, take that. Once you are on the new server then you will be able to pay to transfer to Whitemane as long as Whitemane isn’t locked.
Last I checked the servers that were locked were Benediction, Grobbulus and Faerlina.
Edit: Including Eranikus thanks to the poster below me.
Here is the link to the limitations on free transfers: https://us.battle.net/support/en/article/000032067
Specifically: " * You can use the paid transfer service anytime after a free transfer, as the free transfer does not apply a transfer cooldown"
That aside, the customer service you received is pretty poor. In general, it feels almost like we are responding with AI/Bots more than even someone at a contracted call center.
Second Edit: It appears that Whitemane is one of the servers on this list that is locked → Transfer Restrictions and Free Character Transfers Now Available -- Updated Nov. 22
I would push to escalate the issue. Unless this post gets visibility, you aren’t going to get anywhere with this. Not only that but I would post specifically to that forum post if possible. I guarantee you aren’t the only one impacted and like with most things, the squeeky wheel gets the grease.
The number of accounts associated with characters on a realm IS the issue, not the number of characters. If you have a character on Grobbulus, you can create as many as 49 other characters (or however many the cap is) right now and there is nothing stopping you.
I cant speak to the limitations here but it seems that the transfers work on a “is server open or closed” basis and just stops right there. Any detail or nuance that you provided to those reps was clearly lost.