Hello!
I did a similar "addition" on an interior door - though not a closet door.
I used 3 or 4 mm Masonite, glued and nailed it with a nail gun.
Along the edges, I applied a strip of latex caulking to fill the gap and then patched and sanded.
It has held up for two years without any cracks.
MDF board in 6mm thickness can be found at hardware stores, or they can order it for you.
If you go to someone with their own saw, you can ask to have them cut into 5 or 6 cm strips.
Keep in mind that even with MDF it can become heavy, depending on how much you attach...
On the other hand, I have thick lacquered closet doors with routed mirrors from Ballingslöv, which are MDF and they hang well on three hinges each... d^_^b
Spontaneously, I thought about how the doors "behave" when you open them...
If you place the closets right next to each other, make sure the doors don't touch the next door when you attach moldings along the outer edges.
Best regards,
Daniel