1 by itself is tough. Starting pen is good for just that, starting a pup on rabbits, but after that you need to get the pup with dogs that know what they’re doing.
As for running off game, the shock collar is your friend. The dog is going to run a deer eventually, just accept it. When they do you’ll want the collar turned on and the transmitter in your hand.....that’s how you break them from running deer.
While you’re at it with the dog in the starting pen go ahead and put the collar on it now. Get the dog used to having the collar on every time it hits the ground to hunt. Also, use this time to train the dog to come to you using the tone feature on the collar. Call the dog and hit the tone button. If the dog is on the ground have the collar on it and use the tone every time you call the dog, period. Eventually you’ll get the pup coming to when simply by hitting the tone button and calling for it. You’ll thank me when you need to call the dog in at the end of a hunt.