I think if your flat's internal temperature is 160F your need to keep it in the cooker(foil wrapped at this stage) until it's closer to 200F, when it should be tender, then rest it in the cooler.
I usually prep the cooler with hot water, dump the water, then wrap the foiled brisket ina towel I've pre-heated in the clothes dryer. Let it rest for an hour or 2 , at least.
You mentioned Weber - are you using a kettle or a WSM?
The last 2 briskets I did, I let them go to 205F. I had them cooking in a disposable foil pan so I could keep the juice and drizzle it over the sliced brisket when serving.
Welcome to the forum and good luck with your cook. It's nice to see another Torontonian on board.