I cannot claim my children on my taxes, but have joint custody with my ex-wife. Am I allowed to use my HSA to pay my children's medical expenses and be reimbursed?

Yes.  You can still use your HSA to cover their medical expenses.

According to Pub 969 Qualified medical expenses are those incurred by the
following persons.
1. You and your spouse.
2. All dependents you claim on your tax return.
3. Any person you could have claimed as a dependent
on your return except that:
a. The person filed a joint return;
b. The person had gross income of $5,050 or more;
or
c. You, or your spouse if filing jointly, could be
claimed as a dependent on someone else’s 2024
return.
For this purpose, a child of parents that are divorced, separated, or living apart for the last 6
months of the calendar year is treated as the dependent of both parents whether or not the custodial parent releases the claim to the child’s exemption

So, even though I cannot claim my kids as dependents the IRS sees me as their Dad and I am able to say I could claim them?

3. Any person you could have claimed as a dependent
on your return

So, the IRS would not audit me for paying my kid's medical expenses with my HSA even if she claims them on her taxes and the kids are not on my medical insurance? 

Correct, as long as you lived apart for the last 6 months of the year, for the purpose of the HSA, even if she claims them they are still treated as dependents of BOTH of you.  

There is never a guarantee that you won't be audited for any reason, however, you would not be in the wrong for paying the kids medical expenses with your HSA even though she claims them and you do not have medical insurance on them.