Morning is a great time to visit the palaces. Gyeongbok is the larger of the two palaces and a visit there can easily take the whole morning. The
Changing of the Royal Guards at Gyeongbok is a colorful ceremony not to be missed. Also inside is a place where you can dress up in traditional Korean outfits and have pictures taken around the palace.
Changdeok is a bit smaller, and if i remember correctly, you can only take a guided tour of the palace. At least at Gyeongbok you can go pretty much anywhere you want to. But with
Changdeok it is guided tour only. This might have changed but that is what I remember. Still it might be worth a visit as a guided tour might give much more insight.
You can probably visit both in a morning, but I would pick one and enjoy it. Everyone's style of touring is a bit different so it's hard for me to say what is better.
Also, between Gyeongbok and Changdeok palace I believe is the Hanok village area. Lots of traditional style homes. Also some fun places to eat and shop!
As for bbq in Insadong, I haven't had traditional bbq there, but perhaps
Min's Club might work?
But if you are looking for bbq, Bonga restaurant can also be a good option. They specialize in
WooSamGyup, which is a thinly sliced marinated beef. They have instructions on how to cook and eat on the previous link. Plus you get a lot of fresh vegetables to wrap the meat and it makes for a delicious meal. In addition to WooSamGyup, they also have various other
meat as well. The
original location is open 24 hours and is located between Nonhyeon station and Sinnonhyeon station on a back alley famous for all kinds of different restaurants. A bit crazy at night. But if you go to this location, take a taxi there and be prepared for a long wait as it is very popular. However, they have multiple locations and there is one located
near Insadong. Easily walking distance from the south end of Insadong. You will have to cross a big street and then navigate into a slightly crowded and busy area during night time. Lunch time will probably be fine, but at night be prepare for the flashing lights, music and probably a few soju'd businessmen. Still very safe, it just adds to the scenery.