What time of the year are you visiting ? If it's during the warmer summer season, then I would definitely go for the HDDR - its inside vs. outside.
I would probably lean more toward the HDDR anyway, if you've never been to either. . . . A two year old would definitely be able to sit through it - its pretty fast paced and well orchestrated. Although I would try for the earliest show (5pm-ish) if you wanted him to stay awake throughout the whole show. I've taken 2-yr olds several times (including triplets, what was I thinking !?!?), and they've all done well for the most part. I had to go outside for a brief walk/bounce once, but thats about it . . . . .
Mickeys Backyard is fun also, however there tends to be a rush of parents/children approaching the characters, which can be kind of tough on tired parents and 2-yr olds - maybe in a few years ?
We did Mickey's BBQ with ourn daughter a few years ago, and although the food was good, if you are wanting character interaction, I would try another type of meal... It was crazy with kids running all over the place, and since there was no set line to intermingle with the characters, they just come out and stay in the middle of the stage area, I'm sure you can imagine what it was like trying to meet them and take pictures... You sit at long picnic tables, so unless you sit across from each other, you can't even talk really well to the other members of your party... If you aren't realy big on trying to get pictures with the characters, and don't mind a disorganized setting, then your kids may like it, since there is lots of music and the kids can run around and dance when ever they want too... It is outside, but under a covered area, so if it rains you're still ok... As I said earlier, the food was good, but I did not like how the character interaction went...