In my opinion, I don't think your bulldog is just being picky. I think the food switch unsettled his stomach and he's still trying to recover. I suggest to stick with the food that works and prepare extra food in advance. The way I do things, I always have an extra bag of dog food (well, I have more than one extra bag because it's also part of my emergency preparedness system, like if a hurricane hits and the stores are closed for weeks). So, I buy new dog food when I'm about to use the extra bag. This works well if you use Petflow (somewhere on this site is a code for free shipping) because you can order your new food once you open the last bag of dog food so you'll have a good cushion of when your new bag gets to you in the mail and when the old bag is used up.
In the meantime, I suggest to give your dog a 24-hour no food, not even treats, fasting period to settle his stomach. Then slowly introduce his food again starting with half his normal serving for the first day then working your way up to his full meal the succeeding 2 days. If this doesn't work, you might have to go to a bland diet to get your bulldog's tummy to settle down. There are several "homemade recipes" for a bland diet on this site somewhere.
Hope this helps.