Honestly, that will be really tight. This event has a show not just a plain character meal. And let me also stress, the characters do NOT come to you in this venue. You must get up and go to them to engage. The whole thing took us more than an hour and my nephew was a little older than most of the kids there. FW is pretty far from AK. If you are using your own transportation, you have to get from the back of FW to the front (either over a mile walk or an internal bus which can take over 20 min) and then drive the 15 min to AK and then park, walk to ROL.
Using WDW transportation, either an internal bus (can take 20 min just to the front of FW) to the main bus stop at FW then wait for another bus to AK. This could literally take 4O min or more just for the bus. Using water taxi to MK to AK could take just as long.
In MBYBBQ at 5:30 - about an hour for food/show maybe leaving between 6:30 - 7 (with rushing everything) and then calculating in another 40 min for transport, you'd probably be exahausted from running and sick cause you just ate a big meal, not get good seats and barely make it for the beginning of ROL. I always count on 40 min to an hour for transportation around WDW. Even if the transportation faries sprinkle you with their pixie dust, you really are cutting it close.
Even using the new Minnie Lyft option won't save you that much more time. Maybe 15 min.
And if you and your family are having fun at the BBQ do you want to rush them away from a character experience? Many people lingered when we did this a few years back to grab dessert or their kids wanted to play on the playground right outside the pavillion.
If you want to do something after the BBQ, I'd head to the main pool area at FW and do Chip & Dale's sing along camp fire & movie.