This was a bit of a long experiment, but here are the results:
I tested for about a week with both the old and new trackpads together, and everything worked well.
I tested with the old trackpad on its own, and all was well.
I then went back to the new trackpad on its own. As soon as I connected it, TotalSpaces stopped receiving the four-finger drag event. I restarted TotalSpaces, and then for about a week it seemed to work well, and suddenly there was a short break in communication between the trackpad and the Mac, and TotalSpaces immediately stopped receiving the event.
An interesting note: at this point I did not restart TotalSpaces, but I turned on the old trackpad to make sure that it worked. And as soon as I did that, TotalSpaces started receiving the event from both trackpads, and continued receiving the event from the new trackpad when I switched the old off again.
Now, go figure…