Unless you need the summer grades to help boost your gpa, submit your application and transcripts without those grades. As long as you have everything else in to AADSAS, your application will be considered complete once they have recieved your transcripts and sent out to your schools. If submit your application to AADSAS say in June, but you wait to send your transcripts until August, your application will sit at AADSAS until August.
Once the schools have your AADSAS application and DAT scores they will begin reviewing your application. If you are taking your DAT later in the summer and the schools already have your AADSAS application, your DAT scores will be the only thing they are waiting for and once you take the DAT the scores will be sent to all of your indicated schools, and again your application will be complete. Finally once you finish your summer classes your transcripts can be forwarded to your schools through AADSAS or you can send them directly to each school and they will be added to your application packet.