Why does Paul get annoyed with the slave girl in Acts 16?

Why is this spirit giving seemingly positive witness to Paul's claims?

The same reason Balaam could only bless Israel even though he set out to curse them.

If it is true testimony, why does Paul eventually get annoyed (diaponeo) with the girl?

diaponeo - through the pain, grieved. Paul was pained by her condition. She is a contrast to the quietness of Lydia (meaning 'travil) where the whole household is saved. The demon proclaims Christ but has no sincerity.

   2Ti 3:5 Having a form of godliness, but denying the power thereof: from such turn away.

Paul's response is to show the demon the power of God.

Why didn't he perform the exorcism immediately upon encountering her?

Luke presents the two women as a contrast. One who quietly travails and her household is saved, and one who outwardly "prays to be seen of men" (figuratively). The delay makes the parallel. Paul stayed with Lydia and the damsel stayed with Paul.

Just as Acts 12 has a hidden picture of Christ in the events surrounding Peter's escape from prison, Paul may have had a small "dinner theater" in mind as a teaching tool. He let the two grow together (Lydia and the damsel) until the 'judgement'. He modeled God's long-suffering patience in the process.

Even God's patience has a limit.