I have had to do the same, haul into arenas, and I have found that the busier barns require you to pre-schedule your hour and you pay by the hour and if you used the arena after dark and the lights were needed the price was higher. Some places was just a day use fee or a per a horse day fee. I have found most fairgrounds do it that way. Some barn owners I got to know and I just paid a flat monthly fee.
I got to know the fair manager when we lived in Texas and I paid $50 a month and had full run of the place, two indoor arenas and one outdoor, a trail course, wash racks and stalls.
I would call and talk the barn manager, I have had better luck with phone calls or speaking in person rather than emails. Good luck :)
I DON'T LEAD 'EM AND FEED 'EM, I RIDE 'EM AND SLIDE 'EM.