Embraer Captain reported the stall protection system reacted in an unusual manner after a minor wake turbulence encounter.

Narrative

On arrival to JFK in a non 5G filter equipped on the GPS approach (non RNP) to runway 31; we had an erroneous activation of our stall protection system. Inside the FAF we experienced what appeared to be a minor encounter with wake turbulence; I don't remember who we were following but it wasn't a heavy; maybe an A320. The aircraft acted normal with a small rocking of the wings and slight pitch variation. However; the stall protection system acted quite aggressively; disconnecting the autopilot and shooting the low speed awareness tape up and the PLI (Pitch Limit Indicator) down hitting the shaker for a few clacks. The FO (First Officer) was flying and continued to do so. After this brief encounter we ended up a little high; about 1/4 dot. I instructed the FO to continue the descent following the FPA (Flight Path Angle) of 3 degrees. This would keep us mirroring the glide path but just above it hopefully out of any further wake.The encounter was brief; just a second and the aircraft returned to normal. There was no further turbulence. However; a few seconds later descending below 1500 ft the low speed awareness tape as well as the PLI started again being wildly erratic. They jumped from normal all the way to our airspeed and pitch attitude causing shaker activation. ( the only thing I can equate it to is what they do in severe wind shear when we practice that in the sim.... However after our brief encounter with the presumed wake turbulence flight conditions were smooth).The FO and I evaluated the situation and knowing that the energy state of the aircraft was normal; we were already configured for landing and that it was VFR; we elected to continue the approach after verifying there was no airspeed mismatch and all other indications were normal. In order to keep the stall protection system from activating as it continued to wildly vary both the low speed awareness tape and PLI we added 10 KTS to our approach speed and landed normally. In our judgment taking this unknown error airborne with a go around where the only EICAS message was 'shaker anticipated' was more of a risk than the continued approach. While then extra speed was not accounted for in our landing numbers 13L is 10k feet long more than enough to account for the extra speed. We were much more comfortable continuing the last 1500 ft than increasing power and more importantly pitch in a go around; we definitely didn't want to ride the shaker if we could help it longer than was necessary.After landed we got EICAS messages of ADS prob 1;2;4 fail. We wrote up the discrepancy in the log book.We did not declare an emergency in fact it didn't even cross my mind while in the air our hands were very full trying to evaluate what was actually happening and formulating a plan. Had we gone around I would hope that we would have declared one as we definitely would have been in need of priority handling.