Suppose that a participant's account balance is less than $5,000 at her termination, but when the balance is to paid out a little later in the year, the value is now over $5,000. Can the account still be paid out without consent?

No. See Treas Reg. 1.411(a)-11T©(3).

Use the latest valuation date preceding the distribution to determine if the vested balance is more than $5,000.

I've read the part about the prior valuation date in various reference sources, but I don't actually see the basis for that in the regs. What I see is that the value at the time of distribution should be used.

I suppose that it makes intuitive sense that the value of the investments that would be paid out is the value to be used in determining if the value is over $5,000.

But let's assume that technically the most recent valuation date is appropriate. If the plan document allows "valuation date" to be different times for different purposes, do you think that for purposes of determining mandatory cashouts, valuation date be defined as at time of termination?

Following your logic, then I would say that the plan should define the valuation date for this purpose consistent with the regs.

Therefore, I looked at a prototype document and the proposed language they drafted for the GUST amendments. They have defined the date for determining consent requirements as the value at the time of distribution. Obviously that has not been approved, but is consistent.

You may check with your document sponsor to see how they plan to define this, if it is not already defined.