If you get caught it would be tricky.. Anyway it probably wouldn't help you THAT much since a lot of passing the test is getting used to the school bike, as well as going through the circuit stations. Both of which you won't be able to do in your carpark, on your brother's bike.
6 months should be safe. BTT itself is a few weeks' waiting time. The final test is 1 - 2 months waiting as well. I cleared all the lessons with a single try and easily finished in 6 months. You're probably more limited by waiting time for tests (which should be the same for all schools) than the school schedules themselves. Don't pay too much attention to their estimates