Fin Rot is NOTORIOUSLY nasty, :/ I've had half of a fish's tail disappear in less than a few hours, so I think it's better not to underestimate and ALWAYS take safety precautions.
But to be on the safe side, I'd recommend a 70% water change and add 1tsp of aquarium salt per gallon to the tank. This will help promote fin healing/regrowth and help fight against the possible chance of fin rot. Perhaps even salt dips a couple times every day would help (1tbs in 1 gallon of water for 5 minutes. Then half salt dip water with half tank water for 10 minutes, and then slowly introduce him back into his normal tank.) When adding aquarium salt, MAKE SURE IT'S FULL DISSOLVED before adding fish.
Taking down the construction paper will defiantly help too, if you want to have something on the back, I find white paper sometimes does the trick (that is if the light is in the right place and doesn't cast a shadow on the paper... Bettas sometimes will chase their shadow.) Although turning his light off during treatment can help him not get so stressed out :)
And Stress Coat is NOT a type of medicine, it's a water conditioner, so please be careful not to overdose.
But if possible, can you please take a picture of your fish/fish setup? And if you got water tests (for ammonia, Nitrite and Nitrate. Or take a water sample to a pet store like petsmart and they will test your water for you), please post the results :)