Take the online entrance exam first, pass it, and then contact the exam organizers. Tell them you've already passed the entrance exam. They will tell you that their test is full, and you ask to be put on a waiting list.
Begin Rant:
Many exam organizers are not approaching this correctly. For example, I just took an exam last weekend in Detroit, the organizer contacted me 10 days ago telling me that I had a spot if I wanted it. I had been way down on the waiting list, so that means many who he had given reserved spots to didn't pass the entrance or weren't serious. The end result is I had very little notice but it just happened that I could travel to Detroit that weekend (a five hour drive).
Also, the test was only 3/4s full - 9 out of a maximum of 12. I know, first-hand, how challenging it is for people to get into the tests because of the way the tests are being organized and here's a test that wasn't even close to being full.
Another test organizer I know is doing it right. She has a class and encourages class members to take the exam. But, she will not even write your name down until you pass the entrance exam. Then she methodically goes through the list by category - those who take her class get first shot, current bcjp judges get second shot, others from her local area get third shot, etc.
In addition, she requires you to pay in advance. The upshot is its organized, everyone knows months ahead of time that they will be taking the exam, that they have an assured spot, and she has very few people pull out at the last minute.
End Rant.
That test was proceeded by a class as well (I supplied some of your flawed beers for it.. haha... sorry) I don't believe the issues were from the online exam. I think they were more likely just from people backing out. I do agree taking money up front might reduce this. The sign up lists for his exam fills up as soon as he starts taking names. So most people should know way in advance the date/time etc... But then they are cancelling. I do agree with your suggestions though of requiring payment first. I would also give all those that sign up a deadline on the online exam (maybe 2 months ahead of the exam date). He gives priority on the sign up sheet to those that take his class... so they always get first dibbs...but yeah I would do it a little differently.