The easiest thing for you to do would be to get the other side of that connector coming off of the rectifier, and then run a wire from there to the battery side of the starter solenoid or to the positive battery terminal of the battery with a loop connector on that end of the wire. It looks like the mower had the 6 pin connector so if the new engine has the 6 pin connector they should be just plug and play. I am surprised that the new engine doesn't already have the wire coming from the rectifier to the 6 pin connector.
Something else to check would be to stop by your local Briggs dealer and show them those pictures. They should have what is called a electrical repower kit, which contains all the parts to build the wiring harness to connect to your 6 pin connector on the mower. You may also be able to use parts of the wiring harness off of the old engiine.
Here is what I was thinking of doing before I posted the original comments: One of the views shows the incoming red battery cable connection on the solenoid. It includes a smaller red wire interrupted with a single wire connector. The other end of that wire is connected back into the solenoid, so it appears to me that that wire is feeding power to that solenoid for distribution & that has to stay as a necessary connection. I also believe that that same incoming connection should have another wire coming off of it to go to the single wire coming out of the rectifier. Your thoughts on that.
...
...