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not sure what to do with my toddler pissing off my downstairs neighbor?

ok. so my son is 2 1/2 years old. happy, hyper little boy. we just moved a couple months ago to another apt in the same complex. well the neighbors below us is a handicap old lady that has always been rude to us, ALWAYS. then there is her daughter, her daughters husband/bf/baby daddy(?) and son, plus random people in and out all the time. we have no issues with anyone else in the apartment but the handicap woman. she uses a can and also her electric wheelchair sometimes and uses an air tank. we hear her yelling ALL THE TIME. so my son, being a 2 1/5 year old gets excited and runs around and stomps, when he gets mad/frustrated/throws a tantrum he will kick/stomp/hit/scratch... he stomps more b/c he knows it will get a rise out of us. so when he makes any noise she uses her cane and bangs on her ceiling/our floor. so then he thinks its a game and stomps back. she is the only one in the apartment with an issue btw. so on 4th of july we got back from DC around 11pm, we were in the door for 5 minutes. i got in the shower, i tried to get my son to come in with me(he normally LOVES baths and showers) but he threw a tantrum, so i had my bf come in and get him out of the bathroom to calm him down. the woman (some how quickly being that she is handicap) came up the stairs within the 5 minutes of us being in the door, starting banging our door with her cane, my bf answered and she yelled at him that she was going to call the cops if we didnt keep him quiet. now i get home late from work bc i normally close at starbucks. so i get home around 9:30-11pm depending on when i get out of work. my son gets excited bc im finally home, so i starts running everywhere and wont go to bed without a fight. so i have 2 options, let him stay up till he eventually passes out and try to keep him happy by giving him whatever he wants so he doesnt throw a tantrum. or i can try to get him to bed early and then hes gonna throw a tantrum and stomp and scream. so tonight my son was a little excited, wasnt even as noisy as he usually is. there was a heat and air quality advisory so he didnt get to go play outside all day. i get an officer at my door tonight around midnight bc she called crying and complaining about my son being too loud. he didnt even want to hear me explain anything, he just said he will fine us 0 if he keeps having to come out. WHAT AM I SUPPOSE TO DO???? i feel like im suppose to just let my son get and have whatever he wants just to make this mean and crazy lady happy. i dont want to move again since we just moved (we needed more space thats why we moved). btw, i found out from another neighbor that the people in our apt before us moved bc of the woman downstairs. and we have already talked to our rental office about her in writing 2 times and once in person, but all they can do is talk to her. please... i need any and all GOOD advice. thanks!
so *he* gets excited i mean.
btw, i really do pick my battles with my son. if he wants to fall asleep with the lights on i let him, if he wants chocolate milk(not the syrup, i use powder so it has less sugar and more vitamins) instead of white milk, i let him, if he wants to eat peanut butter sandwiches 4 meals in a row i let him. im really lenient with him except for health and safety issues and his stomping. also, i have always been nice to her, shes met my son, she has always been rude to us and "barks" (figure of speech, shes not really barking) at us when we try to be nice to her. and she wont acknowledge us unless she wants to complain.
oh and when we do correct him, we try to stick to our guns about the correction (figure of speech, no actually guns are involved)