My 2010 developed the dreaded brake issue

I have a 2010 (purchased used in 2013 with 18k miles) that has just crossed over 200k miles.

2 weeks ago, while coming out of a parking spot, my brakes went all weird, almost causing me to hit the car behind me while reversing, and the usual warning lights (Brake, VSC, ABS) lit up. Drove home and, parked and took to Google where I learned about the known problems with the ABS system. Coincidentally, yesterday my car refused to start, so I purchased/installed a new 12V Optima Yellow Top battery - (I was surprised that the original battery manufactured in 04/2010 made it this far!). I had hopes that the failing 12V was the cause of the brake pump issue, but it still remains.

I have discovered, quite accidentally, that if I press the brake pedal to the floor and hold it there for about 20-30 seconds, the warning lights disappear and braking function/feel returns to normal. This will last for about 10 miles or so until the lights back come on again. Once this happens, I can hold the pedal to the floor again to clear the lights. Why does this happen? Should note that even when the lights clear and the brake again feels normal, I can still hear the pump off off every 10 seconds or so.

I am now at the point where I need to decide whether to take the car in for diagnosis/repair or just move on from this car. It's been a great car with no major problems until now, and I always told myself I just want to make it to 200k miles, but I fear other problems may start popping up at this age/mileage. Any general advice on anything I can try - I'm most curious about why I can clear the issue by flooring the brake pedal. Also, I am only "moderately ok" with car repair so this doesn't seem like a job I would take on myself if replacement is necessary. Thanks for any help

 

I posted in the wrong forum. My car is a Gen 3 Hatchback

 

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