I'm confident with fixing the new skin down to the frame it's just whether or not causing the skin to bow is easily done?
When i was taught to skin a door, i was told if you do it right, no filler is needed. if you find yourself using filler, you have done it wrong.
When folding the skin over, you need to first lay the frame onto the skin so the door is skin side down. It needs to be well supported so it's not rocking around all over the place as you fold.
I always fold just the corners first to hold the skin in place and then try the door on the car. if the skin is in the wrong place, it can easily be moved.
I would advise not using any 'door skinning' tools. They don't make a better or quicker job in my opinion. best bet is a hammer and dolly. You want a dolly with a nice weight to hold against the outside of the skin where you are folding. you need to hold it with enough pressure to support the skin but not to bounce around while you are hammering.
use a bumping panel hammer and carefully make your way round the skin knocking the edge over a 1/3 of the way, then go round again another 1/3 and then a final run to fold it up tight.
If done well and with patience, you will have a nice smooth ripple free finish that is ready to paint.
You could always add a bead of sikaflex around the edge of the skin prior to dropping the frame on and as you hammer the edge over, it will squeeze out and create a decent seal all the way round.
You can also use one of those skinning tools to go round once after the edges are folded just to tighten the folds up nicely but be careful to avoid rippling.
it's also a good idea to counter sink the water drain holes in the frame before fitting the skin to help with drainage.