Wood has seasonal movement and panels usually crack if their edges are trapped from moving by being glued, rather than being inset without glue so they can float. Probably, no matter what you do, there will be some kind of visible flaw as the crack opens and closes slightly depending on weather. If the crack is not huge, the simplest solution might be to use a good grade of latex caulk because it's flexible, and then paint over it. You can smooth latex caulk by wiping it slightly with a moist paper towel right after its laid down until it looks as good as it will get. If you overdo it, you can put on some more caulk immediately and try again.

Panel doors often crack from paint gluing the panels all the way around.
The panels shrnk in the winter as the humidity falls and split.
If you are going to strip them you may be able to free up the panel, but the paint can be deep inteh grooves supporting teh panel and very hard to remove.
You can fix the crack by freeing the panel and using hot glue to put a block of wood on each side of the crack.
Spread glue liberally on the crack and use a shop vac to pull it into the crack. A clamp on the wood blocks will pull the crack closed and hold it until the glue sets.
Pop the blocks loose with a chisel (hot glue is weak in shear).
When you painthte door again, trry to aply a minimum of paint at the edges of the panel.
Painting the panels, allowing them to dry, cleaning up any paint fillet with a rabbet plane and then paint the remainder of the door being careful not to cover the panel-frame jointa and 'glue' it with paint.
It is a real PITA and hard to prevent the paint from fixing the panels.

Katdip,

My idea maybe dismissed for a number of reasons but what about getting some wood veneer, cutting it to the size of the damaged panel and using it as an overlay that could be primed and painted.

Best of luck with the project, rredogg

Another choice could be "Crawfords wood putty" (look at their web site) or bondo as a filler.

Again, best of luck with the project