Pardon my intrusions but if I were you, I would not promote possible electrical code violations by disturbing that service outlet.My controller has a standard 3-prong plug and there's an outlet by the breaker box, so I'm not sure how I could wire it directly to the pump timer, but I'm sure that outlet could be rewired to be controlled by the proper breaker.
I have no doubt, your swcg is wired for 120vac operation. The poor man's solution is to get a heavy-duty outdoor digital timer for less than $20 like this one. Mine is on its 5th yr and still going strong. Syncing the digital timer with your pump run time is more precise compared to the mechanical timer with trippers. I have mine set to power on the swcg 5 mins after the pump started and off 5 mins before the pump shuts off.
And for safety reasons, you need to make sure that the flow switch was installed correctly and functioning properly. You'll get a lot of hits on how to do just that by searching this forum.