The ones I have installed say they must be on the load side of a 240v breaker.
My main concern with the double tap is the size and type of the lead. I doubt the SqD and CH breakers are listed for a #10 solid and trhe smaller stranded you get on the protector. I had the slots so I put mine on a separate 20a double pole.
From a purely engineering protection point-of-view it really doesn’t matter where it is connected as long as the wires are kept relatively short (because of industance as mentioned above) and the device is connected to both busses (i.e. via the SE lugs or 2 pole breaker). Note that curling the long wires into a coil to keep them “short” would about the worst thing to do.
The main reason that the usual recommendation is to connect thorough a breaker is because often, when a surge arrestor does its job (or tries to), it typically fails at the same time and will need to be replaced. Most of the time I would expect it to fail “clear”, thus the need to be able to see the light. If the failure mode is really severe, it will trip that breaker. A connection at the SE lugs would mean having to de-energize the entire panel, often with the assistance of the local utility.