What is the proper way to cook a hamburger on a Pit Barrel Cooker that doesn't result in fast cooking taste?
There is no method because you cannot do it. Ground meats lose moisture rapidly when cooked as much of the fat is not contained within the meat itself, but mixed in. This mixed in fat melts rapidly and runs out. The fat that is marbled within the meat itself is lost faster as the ground strands have much more surface area. Therefore, if you cook a burger low and slow it will end up dry. 

Slow cooking and long roasting is only good when a piece of meat has enough marbled fat so it will still be juicy when cooked, and a low surface area to slow moisture loss (ie it's fairly big). Burger cooking on the other hand is a race to get a good crust and to the right temperature before losing all moisture. 

As for shrinking up this is normal for burgers and you cannot prevent the process from happening. You can, however, shape them flatter and thinner to compensate. I make a flat, thin-ish burger patty, then I use my palm to make the middle slightly thinner than the outside. The result is a burger that has the right shape after cooking. If you start with a thick burger patty you'll more often end up with a meatball shape at the end.