I don't know if you have completed your project but here I go !
This is just my opinion, you will need to make any final decisions.
Just about any Screw of proper length will work, it all depends on what look you want; me I would go with the an Allen Head, or a Torx Head, or even one of the Security Torx Heads, something that would stand out. Get a thread that matches you existing screw may make it easier but the head of the screw will determine what you use. There are on line stores that handle nothing but Fasteners.
Looking at the panel size, I am thinking you will need more than just 4 screws in the corners, maybe start with the OEM mounting positions, then measure between them and position mounting points using the "Middle Technique"; I know WTF is that !
What I do is, using the corner screws as a starting point, find the middle of those screws and mark a hole for that mounting point, then find the middle of the those and so on and so on. You will need to have an idea of how many will look right for your truck AND what style you want but it will give a more linear look, also... you may need to follow the lines of the door, so keep it in mind. Make sure the screws on the opposite side / bottom need to be closely aligned for a good look. More screws will give you better support. The door handles and arm rest should hold the center in place but a mounting point in the middle of the door may be needed.
Nothing says you need to keep the same size screws, you could go larger so you can have a larger head to hold things in place.
Also, a machine crew can sometimes be used in the Door Clip sale of panel mounting, something to think on.
AGAIN, these are just an ideas, best to draw something out first... AND if possible find a large piece of cardboard, LARGER THAN THE DOOR PANEL, and make a panel out of Cardboard, then start by just drawing the mounting holes and placing it on the door to see what you want.
Then you can either transfer the holes or as I would do, tape up the Wooden Door Panel and remeasure everything, TWICE !
Oh ya, do not drill the door until you have first decided on the best mounting screw setup AND second, make sure nothing is in the way of the new mounting holes and hardware.
OK I'm done, good luck with your project, take you time and do it right the first time !