529 distribution pro-rata with multiple accounts

Does the beneficiary designation on any of the accounts affect the tax treatment?

Yes it does. If child 1 has $2,000 in qualified educational expenses, and child 2 has zero qualified educational expenses. Then any money pulled out of an account with child 2 as the beneficiary could face taxes and penalties.

If the account with child 1 is running out of money there is a method to transfer money between the funds. This change of beneficiary doesn't cause any tax issues.

This whole issue in your question gets more complex because how it is reflected in the tax forms depends on if the funds were transferred to you the parent, or directly to the school.

When I had two children in college, I had the money sent to me instead of the school because then I didn't have to worry about what happens if my child drops a class, and gets a refund. The problem is that since the 529 sent the money to me, the tax form from the 529 at the end of the year combined all the withdrawals for both children onto one tax form. I had to pull them apart to correctly document that the qualified educational expenses exceeded the withdrawals.

In the example in your question, the money taken from account #2 will be part contributions and part gains. There is no way to pull only contributions from the account. Now if the account is so new that there are no gains, then it will be 100% contributions.