RE: Need some advice/help on electrical work on PT 109
hansvonpensch is 100% correct. Get rid of the old ESC. It is a mechanical type and may no longer work very well. The Proboat is fine and the Viper brand is also a good choice. As for "go power", it is a bit tough to get jaw dropping performance on one small prop and one small motor. If the performance is not up to par, you may want to go with larger components (prop, motor, etc) or convert to a twin prop setup. The two spare battery clips appear to be for the main battery pack. You can use the type used in rc cars, a 6cell 3000mAH Nickel Metal pack but you'll need to change out the clips for Tamiya connectors. You'll also need a battery charger.
Also check with the local hobby store on the vintage of your radio. If this is really on old boat, you may not have a legal radio. With a lot of folks now converting to 2.4Ghz, you may find numerous 75mhz radios available at local hobby shows and swap meets at a really great price.
Is there a ship model club in the area? They would be glad to help out with your project. You can work through these issues on your own (with our help), but it would be best if you had a crew around you that had been down this road before.