We have done both with two younger kids (now 9 & 7 but younger then). Both kids and adults enjoyed each experience very much. The food and the show at Hoop-Dee-Do are much better IMO.
The Backyard BBQ is fun, but there are no servers and you miss a lot of the show waiting in line to get food and drinks. The audience participation is great at this one as well. If it still isn't too hot every one can have a blast with the wild west show and dancing.
One thing to consider (for either experience) with two kiddos is to go earlier in your stay. We made the mistake of going to Hoop-Dee-Do on our second to last day during our summer '15 visit. The kids were so worn out that they were resting their heads on the table, not eating much, and not really feeling it overall. Contrast this to our first visit to Hoop in '14 on our first night during that trip...the kids were singing along, laughing, and really getting into it.