There are lots of good options here... but my first idea was to splice in a new piece of aluminum on the seat bracket to make it wider (front to back). you can cut AL with any carbide tool, (skill saw,etc.) so cut it in half then get another piece from a metal supplier and drill and bolt it together so that it fits. if you think ahead you could probably even use the seat bolts to hold it together... This idea may work and would be my first choice if the seat will drop all the way down, but that appears to be close (you may have to lower cooler by an inch or so).
You could also just cut off the back "leg" and allow the bracket to set on the cooler.
As far as codycleves suggestion, that's what I've done for years. It works great but there are a few drawbacks... you can't row without the cooler/drybox/whatever. That's the beauty of those brackets, nothing else is needed for support. You can ditch the large cooler for a small one on day trips to save weight and not change any set ups. The other issue with seats on plywood is that they can get left on rocks at the take out...how do I know you ask? Just left my brand new seat on the Grand Rhonde on thursday. Didn't realize it until I was 200 miles away
If it's bolted to the boat it's harder to leave behind...