I believe I am correct in saying that a 12 mm triple start will just do it, with a 10 mm section thread ... you could put out a question specifically on this point. . Or you could research some of the similar pen designs in the Show Off Your Pens Forum.
A 13 mm triple start cap to body thread will definitely do it.
For 12 mm, the pitch is usually 0.75 or 0.8 mm. Then the 10 mm section has 10 mm size as the o.d. of the thread (and this would not be t.s.).
A rough rule, for this 12 mm t.s. thread would be that the depth of the thread is very close to the pitch.
So ... 12 mm (o.d. of body thread) - 2 x 0.8 mm (once for each end of the diameter) = 10.4 mm
Subtracting from this the o.d. of the section thread: 10.4 mm - 10 mm = 0.4 mm
Then you have to divide this by 2 to get the "wall thickness" between the two sets of threads = 0.2 mm.
That is really awfully thin and I would be inclined to go for 13 mm t.s. as the cap to body thread.
I don't have as much experience as many others, but that calc. is only intended to give you the idea.
Of course, you have to also consider the overall diameter of the pen (14 mm) and the fact that you have to tap the cap also. . If you go for a 13 mm cap to body, you are going to have a very thin "wall thickness" under the 14 mm overall o.d. . (14 mm - 13 mm) divided by 2 = 0.5 mm and that is OK for the 13 mm choice. . If you do try for a 12 mm cap to body, then you get the same result if you go down to a 13 mm overall pen o.d.
I know that tap and die sets are expensive. . So it would be good if you could borrow a 12 mm tap/die set and give it a try. . It just might work out.
The fact that the 12 mm is triple start whereas the underlying 10 mm is not may work in your favor.
If the pitch is 0.75 mm instead of 0.8 mm that will also work in your favor.
It might be possible to go to a section thread of 9.5 mm instead of 10 mm and that would work.
If I have made any gross error, I hope an expert will correct me.