page 20 of the KPA500 manual shows how to connect up the amp and tuner.
You do need the right cables as designated in the table.
the DIN cable picks up the PTT line on the 7300 and the analog band voltage so the amp knows which band the radio is on. It looks like Elecraft does sell the cable From the 13 pin DIN connector so you would have to build one for the band switching.
The 4 pin flat white connector carries the Tune indication to the tuner, along with the power.
I don't have the combination, but I have been working on a similar set up for a friend with a flex radio, and I’m familiar with the 7300.