You missed the deadline for apply for HS exemption but can file a delinquent exemption for up to two years.
You do not want to protest an appraised value that is less than purchase price. You certainly are not going to get it lowered and they may be able to raise (extremely rare).
Edited to Add: On our old house the square footage of the house was inexplicably lowered in year 3 of the 15 years we were in the house. House is 2495sf and even after we sold it is still at 2106. Advice from multiple people was to not let them know and use it as a selling point. We did provide an independent confirmation of the square footage to buyer.
"The absence of the word accountability is not the same as wanting no accountability" -unknown
"You can never go wrong by staying silent if there is nothing apt to say" -Walter Isaacson