This is based on my very quick reading of the studies so take with grain of salt. The NBER study (OP) studied the entire fast food worker industry using data from BLS, whereas the Berkley study cautions against using BLS because it applies to the entire industry. The $20 minimum wage requirement only applied to fast food workers who work at limited service restaurants with 60 or more chains.
If your concern is only for who the $20 minimum wage was supposed to affect, then there was likely no decrease in jobs based on only that data. However, since causes have effects on more than one intended group, it's very likely that the $20 increase did reduce employment overall and the Berkley study was very careful to downplay that data as not being useful for the purposes of their study, even though they are related. The effects on one part of the industry can affect the rest and to ignore it is a questionable choice.
A hamburger is at McDonalds is $1.89 in my area, a McDouble is $3.29. The double cheeseburger is $3.99. What they call the "daily double" which is a silly name for a hamburger with the works is also $3.99.
I don't think using the basic burgers is a bad choice since specialty burgers probably don't compare well across chains.
