Regarding The Best of Times; I don't think they'd mind playing it live in general, at one point or another; after all, musically, this is by no means "MP's baby" like the SDV/Kevin Moore dilemma. This song is very DT, you can hear JP's and even JR's influence on its writing. Lyrically, singing these kind words about his dad would be more of an honor and a tribute, not a personal offense or whatever.
Playing it for the first time ever in a live setting when you're shooting a DVD, however, is almost suicidal in some perspective. I know they can rehearse offstage and whatnot, but I can't imagine it going smoothly without something fucking up. Some of JLB's playback-backing vocals would not fit right, or he'll have to rerecord something in the studio or Mangini would not be as well prepared as he is on the other songs. Seems to risky to me.