Why not just go to the gate agent or customer service and request a new boarding pass? Say that the old one got mangled, coffee spilled on it, flushed down the toilet accidentally, etc. Not at all unusual to do so; I probably only board the flight with the same BP I used to clear TSA about a third of the time. (combination of upgrades, seat changes, that I tend to print multiple BPs from the online system in case I misplace one, etc.)

Will 1) the new BP have SSSS on it, and 2) even if it does will the GA require the pax to get rescreened?

If both (1) and (2), it would seem that any SSSS pax who has their seat changed post-TSA is pretty much hosed.