Will my ex-husband, who is the custodian of our daughters 529, have to pay taxes on the earnings of the 529 since our daughter is not listed as his dependent?

It’s complicated. You will need to coordinate with him.

For 529 plans, there is an “owner” (usually the parent), and a “beneficiary” (usually the student dependent). The "recipient" of the distribution can be either the owner or the beneficiary depending on who the money was sent to. When the money goes directly from the Qualified Tuition Plan (QTP) to the school, the student is the "recipient". The distribution will be reported on IRS form 1099-Q. 
The 1099-Q gets reported on the recipient's return.** The recipient's name & SS# will be on the 1099-Q.
Even though the 1099-Q is going on the student's  or non custodial parent's return, the 1098-T should go on  your  return, so you can claim the education credit. You can do this because he is your dependent.

You can and should claim the tuition credit before the other parent (or student recipient) claims the 529 plan earnings exclusion. The educational expenses he claims for the 1099-Q should be reduced by the amount of educational expenses you claim for the credit.
But be aware, you can not double dip. He cannot count the same tuition money that gets you tuition credit to get exclusion from the taxability of the earnings (interest) on the 529 plan. Since the credit is more generous; use as much of the tuition as is needed for the credit and the rest for the interest exclusion. Another special rule allows you to claim the tuition credit even though it was "his" money that paid the tuition.
In addition, there is another rule that says the 10% penalty is waived if he was unable to cover the 529 plan withdrawal with educational expenses either because he got scholarships or the expenses were used (by him or the parent) to claim the credits. He'll have to pay tax on the earnings,  but not the penalty.

Total qualified expenses (including room & board) less amounts paid by scholarship less amounts used to claim the Tuition credit equals the amount you can use to claim the earnings exclusion on the 1099-Q. 
Example:
  $10,000 in educational expenses(including room & board)

   -$3000 paid by tax free scholarship

   -$4000 used to claim the American Opportunity credit

 =$3000 Can be used against the 1099-Q on the other parent's (or student's) return

Box 1 of the 1099-Q is $5000

Box 2 is $600

3000/5000=60% of the earnings are tax free

60%x600= $360

He has $240 of taxable income (600-360)

**Alternatively; he can just not report the 1099-Q, at all, if your student-beneficiary has sufficient educational expenses, including room & board (even if he lives at home) to cover the distribution. You would still have to do the math to see if there were enough expenses left over for you to claim the tuition credit. Again, you cannot double dip!  When the box 1 amount on form 1099-Q is fully covered by expenses, TurboTax will enter nothing about the 1099-Q on the actual tax forms. It will prepare a 1099-Q worksheet for your records, in case of an IRS inquiry.