What size thread for the cap is a difficult question to answer Alan, as it relies on so many other things.
Before you can arrive at the answer you first have to consider the requirement to get an ink converter in and out of the pen, which requires a minimum clearance of 8mm, and preferably a tad more. It comes out through the middle of the thread that holds the section in place. You probably already know that to calculate the hole you drill in order to tap a thread, (the minor diameter) you subtract the pitch of the thread from its major diameter - therefore for instance, if you used a thread of say 9x1 you would need to drill at 8mm. That would give you the clearance you need for the converter, so all well and good. 9x0.75 or x0.7 would be even better since the thread would be a bit finer and the clearance would be a bit greater.
So now you've got a hole of at least 8mm and a thread of, we've said 9mm. The size of that thread however could be greater, depending on your design, but I'll stick with 9 for the purposes of this explanation.
The material the that the thread is cut in to though, has a thickness - the wall on the outside of the thread, which could easily vary depending on the size of the pen, the size of the nib you want to use, and the medium you're using. If you assume though that the wall is 1mm thick all round, that adds another 2mm to the 9mm you already have, so you're up to 11mm. It could be a lot more than that if you're designing a fat pen, or less if you're designing a thin pen and using something like aluminium to make it from.
If you go with 11mm for now though, and assume that you're putting your cap thread directly over your section connector thread, that 11mm represents the minimum size of the minor diameter of the cap thread - the diameter of the troughs of the male cap thread. The minor is, as I've already mentioned, major minus pitch, so in this instance, 12x0.8 would do the job, and you'd actually size the thing to 12 instead of 11mm. (in fact a shade under 12, but that's a different matter for another time)
The best thing to do is to sit down with paper and pencil, an eraser and your favourite dram, and start working out how you want it to look, and the geometry necessary to achieve it. Knock up some prototypes using what you've got or what you can afford to buy on the understanding that you might have bought the wrong thing, then once you've got it all sorted both in theory and in practice, invest in the size of triple lead taps you've landed on. It probably will be 12x0.8 - that's certainly the most popular size, but you might find that you want to go for something else for whatever reason.
I hope that helps
Edit: Yes, the Axi MT die holder will do the job as good as any. Unless you're going for a 14mm thread, which I doubt, all our triple lead dies are 1 inch diameter. The 14mm is 1.5" but the Axi unit will hold that one too