Now it's asking me to send some btc from an address I control, what a hassle! I expected to avoid that by having my own Byteball address

Having your
own byteball address isn't enough to receive blackbytes, you need it to be linked to a bitcoin address. For that you are not obliged to send a micropayment, you can sign your byteball address with your bitcoin address, and give the signature to the bot to link both addresses.