These are both fun venues, but for your group I'd choose Chef Mickey's here's why ...
Mickey's Backyard BBQ takes place in a covered, but still outdoor, pavilion. Most of the time in May this should not be a problem, but should you be at WDW during a particularly sweltering day, the BBQ area could be very hot. Disney does it's best with fans, etc., but there's only so cool you can make the outdoors. If you have temperature sensitive senior citizens in your group, this is something to consider.
Unlike at the other character dining experiences, at the BBQ the characters do not come around to each table. Instead, they are out on the dance floor and guests are invited to walk up to them and join them to dance. This generally works well with brave kids ages about 3 to 8. The smallest children would be left out of the action, and older kids might be "too cool" to get up and dance with Mickey. At other character meals, including Chef Mickey's, the characters will greet every guest at every table. You can get photos here of your baby with Mickey that you will cherish later. It will also be MUCH easier at Chef Mickey's to corral your entire party together for a group photo at the table.
The menu items at the BBQ are limited to ribs, chicken, and hot dogs, with sides. At Chef Mickey's there is a full buffet. Picky eaters have many more choices to suit their palates.
The Backyard BBQ can be lots of fun, but for a multigenerational group, I'd go with Chef Mickey's.