Not all the Westerbeke's are fitted with these relays, I believe they started in late 90's. They use these as a pilot relay to power the fuel solenoid and as a hold-in relay.
I've had the same symptoms you're describing, replacing the relay pointed to took care of it, had the issue more than once with the relay. If the contacts in the relay become worn, the fuel solenoid doesn't get full voltage and the engine bogs until the relay contacts make good contact and the fuel solenoid pulls in fully.
They're inexpensive, and they DO NOT have to be OEM Westerbeke parts which would make them about 3X the cost. There are 2 of them in the control box mounted above the generator. I keep a couple of these in spare inventory. Available at any auto parts place.
Could be your problem. Also check the continuity in the 3 safety switches located above the start and preheat switches in the diagram, since any voltage drop through that circuit will create a similar problem. The ladder diagram is very helpful in troubleshooting.