I charge a flat rate.. any accompanist worth hiring will be very forthcoming on their rates. In UIL or solo and ensemble there are usually 3 classes of flat rate according to difficulty and Guild too..one lesson for the performance?... do you have recitals with more than one teacher and is that accompanist there anyway? if an accompanist tells you to pay for the time it takes to learn a piece...then, trust me, they suck don't hire them. Especially if it's a commonly performed work. After a while you know them all.
I teach piano privately. In New York state there's something called NYSSMA, basically through the schools kids get options of pieces for any given instrument at different difficulty levels, and perform them for judges and get a grade. A few times piano students of mine also did it for voice, saxophone, etc. and needed an accompanist. It's not particularly severe in the judging - I had a student do voice level 6 (highest level) last year and get a perfect score without breaking a sweat. - And here's an example of one of the pieces I'll be doing with a student this year. Nothing demanding.But I just I got a call from someone who *just* needed an accompanist for NYSSMA, so I was curious how to charge. I have no more info yet ("Oh...you'll have to talk to her mother she handles that stuff she'll get back to you."), but odds are it will be an easier piece. How many rehearsals we'll need I have no idea I have to get more info first. I don't even know what instrument yet. I just wanted to get an idea what the going system was for charging.
apologies I completely misunderstood your question. Flat rate--you can charge for travel and for wait time additionally. NYSSMA sounds like one of those all day event type deals... man they can be such a pain in the butt. I need to have at least 5-6 kids to play for before I sign up for one of those. Now if it has a regional or state level... then you can pretty much name your price on that round.