Particle board is notoriously susceptible to moisture absorption. My guess is that the bottom of you table is not painted green like the top. So the bare board has absorbed moisture and the top has not. The fix you propose would probably help or maybe even nearly eliminate the bulge but, most likely would take more time and effort to accomplish than you think. If money is not tight I would just reuse the hardware and replace the particle board with Baltic Birch plywood. Even the lower quality USA plywood would be just fine for a table tennis top.
If money is an issue, you can fasten your framing members as you described, by counter sinking large head wood screws through the top. Run a bead of Titebond III between the lumber and the bottom of the top. Fill the holes with lightweight spackling compound and repaint the top so the discolorations won't be a distraction during play. You won't be able to collapse the table of you do this of course, and so it will be very difficult to move. Also, if the particles that make up the board have swelled too much, and it doesn't take much, then the glue line isn't going to help at all and it would take screws about 12" apart depending on just how bad of shape the board has become.
I wouldn't even try the fix simply because the plywood wouldn't be that much, and this method of repair would only be a solid, permanent solution with quite a bit of effort and time. So if you play enough to warrant all that effort, just go the plywood route. Plus, rebuilding the table and painting it (bottom AND top) and striping the white borders would be fun, and you would be proud of your project.
If you don't use it often, just play about 100 games one weekend and get really used to the nuances of the convex top . . . . . then you use that to your advantage to destroy the competition.