There are too many reasons, as you have seen, that the first timer could have burned.
Breakers for motors should be sized this way: Breaker size must be rated for a minimum of 125% of the full-load current of the motor it is protecting, rounded up to the next standard breaker size. This is per article 430.32 in the National Electric Code.
Your pump says 15a. That means the breaker just for your pump needs to be 15aX1.25= 18.75amps. The next standard size is 20 amps. Don't worry about the motor for the timeclock, it's load is too small to have to consider in this case.
For standard loads (which your SWG should be, but the actual breaker requirement should be in the instructions) sizing is 80% of maximum rated load of the breaker rounded up to the next standard size. That would be a 15a breaker (next standard size). So, the 20a breaker could feed the pump circuit and timeclock motor (terminal 1 on the T103.) The neutrals for those loads would be attached to the 20a breaker. That is the pump wiring neutral and timeclock (terminal A). Terminal 2 would feed the pump.
The 15a breaker would feed terminal 3 with the neutral for the SWG connected to that breaker. Terminal 4 would feed the SWG. All grounds would be connected together and to the ground bar in the panel.