You can try getting a walk-up reservation. Go to the check-in at BB as soon as they open and put your name in.
You can also try using a dining alert site like mouse dining. There's a free version and a paid version. I've been successful with them (I use the free version, which means you have to be ready to jump on it when you get the alert). You give them your date(s) and breakfast, lunch, or dinner. They will notify you if a spot opens up and you have to grab it fast.
The downside to the free version is that you can only get an email, not a text, which makes it a lot harder to jump on an opening. There's a work-around, though. You can sign up with your cell number with a special ending that allows texting format.
For instance, when you sign up for Mouse Dining, it will ask for your email. Instead of your email, put in your cell number followed by the ending that goes with your carrier as shown in the article I will link below. Say you have verizon. Where Mouse Dining asks for your email, put in yourcellnumber
@vtext.com. Each carrier has its own @ending so check this article to find yours.
https://support.marketing360.com/hc...o-Set-Up-Email-Notifications-as-Text-Messages
Have as much ready to go as possible in case a notification comes through (Disney dining site to book, Mouse Dining to see the time slot and date). Good luck! It sometimes takes me multiple notifications before I get one but I've always managed to get one.