I think the answer here is NO. The leader cannot direct fire as the MG would technically be forming a "new FG" with the other unit(s). For that matter, I think the leader couldn't join in for the MG's second shot because, again, this would be creating a new, larger FG (leader + MG).
7.53 FIRE DIRECTION: A single leader cannot direct more than one weapon/unit per phase unless they are part of the same FG. Hence a squad that elects to use its inherent FP in a different attack than that of the MG it is manning does not get the leadership benefit if given to the MG instead. However, a leader can direct the fire of a MG as many times as the MG can fire, even if he also directed other units as part of a FG in the MG's previous attack. Leader direction used during Defensive First Fire can be used again in Subsequent First Fire, FPF, or Final Fire, but again only for one firing unit/SW or FG—and that unit/SW/FG can only include firers he directed during First Fire; if forming a new FG or using a different SW during that Player Turn, the leader cannot direct its fire (even during FPF). Similarly, a leader may not affect more than one To Hit attempt per fire phase (except for a multiple ROF weapon) regardless of the number of SW the firing unit is eligible to fire. See also 9.4, 10.7, and D6.65.