The cost you pay isn't just dependent on how many bytes you want to send. Each block maxes out at 1MB and there's only 144 blocks a day. If you want your transaction to be confirmed, you have to outbid everyone else who wants their transaction to happen (as long as there's more than 7 transactions / sec). I don't think it is doable to try to do a transaction with a $3 fee, you need something like $20 worth of BTC to get it done within the next block or two, and maybe $10 if you're willing to wait a while[1]
[1] assuming https://estimatefee.com/ is correct. Based on the transactions of some of my friends it seems pretty accureate though