Persistence. Keep checking online multiple times a day to see if a cancellation comes up. You can continue looking up to the day of if you can't find a reservation in advance. Look for a party of 3 or a party of 4 since parties of 3 are usually seated at a 4 top. The reservation system has a quirk that makes it harder to book an odd number because the system wants to fill all the seats at a table so while a party of 3 will be bypassed, add one person and often there is a better chance to find a table. Everyone needs to be included because of fire code regulations so do not look for a party of 2. If you find a 4, just let the castmember at the podium know there are just 3 of you when you check in and as long as at least one person shows up for the reservation you will not be charged for the "missing" person.
Since you have only a limited number of days be flexible in your time. If you want to meet the Beast or want wine or beer with your meal, you will have to book dinner. If neither of these are important you can also look for a lunch or breakfast reservation. While BOG has a large seating capacity, at dinner not all the rooms or tables are used because they can only seat based on how fast the kitchen can plate meals. Lunch and breakfast are quick service with a more limited menu so there is a faster turnover and more guests can dine.
You can subscribe to the May and June cancellation threads at the top of this sub-forum.
Walking up is a last resort because the week you are going can be busy (Memorial Day weekend, lots of schools are already out).