Had a similar problem on an exterior front door of our Albuquerque house. Except ours was split in 3 or 4 places, both bottom panels. Fixed it with wood filler, thinning it down with lacquer thinner to wedge it into the cracks with a putty knife. Let it cure a few days, sanded it down, and stained to match the door.
A few thoughts on your epoxy idea, as I've used or supervised the use of a lot of epoxies over the years--heating the wood is likely to significantly accelerate the set time of most epoxies, possibly making it harden or at least thicken before flowing into the crack. If you are locked in on epoxy using gravity feed, make sure yours is of low enough viscosity to actually get into the crack. You might consider getting one of the inexpensive injection kits, tape over the crack except for a few small entry openings, and pump away. You hadn't mentioned what the door's finish is, but most epoxies don't take stain very well if that's what you have.