Always best to be careful and go the route you are most comfortable with. To be clear - the word ‘punch’ does sound quite violent however, when working with rollers - and those rollers being made of soft brass - it is best to only exert downwards pressure manually, whilst gently turning it, using an extremely small dome punch or even a pointed punch (in other words - not striking the punch with a hammer). I don't recommend doing this for the first time on a customer owned or otherwise important movement - Its definitely something to practice on though with a donor or scrap movement, it would be a great satisfying skill to learn and add to your arsenal. Gluing is not my favourite way forward, and I won’t recommend it as it is the proverbial kicking the can down the road (hope I don’t sound snobby - I don’t mean to) but I can understand why in certain circumstances this method would be adopted. Absolutely true - this is the biggest risk with attempting to use a staking set to tighten the hole. Very very risky.  Spot on!

From my perspective it doesn't need to be. In this scenario, I'd love some help identifying the correct parts for a conversion. Oscillating weight, bracelet and crystal are not original, so there is margin for adaptations.

I don't understand why it needs to be a split stem. 

Just a small update. The watch has been on a watch stand for the last few days. It has worked correctly, there was only one time when the Date Lever moved out of place. Due to the lack of tightness, the Date Lever lifts a little each time it is activated until it eventually comes off. This process is accelerated when the watch is on the wrist. I hope to have some time soon to resolve the issue Regards

In the other thread you will have read about inserting a hair into the hole of the roller. This would be an easy, non destructive way for a beginner to try affect a repair with a very fine synthetic paint brush thistle. The gluing idea is fiddly but can work "ok" , keep in mind how it might affect what holds the impulse pin in place if heat is required to make alterations to the roller's position afterwards. Maybe I was lucky, but a staking set I bought some years ago had 3 unused sizes of the dicussed stake. It doesn't close the hole, like a dome punch would ( trying that will crush the centre tube section ) but raises 3 notches inside the hole that which then grip the staff. Agreeing with Cees, that it feels somewhat destructive, but it did for me a few months ago. You could actually make your own very easily. Examine the tip of a 4" nail to see the 3  or 4 edges Now using something like a 3mm driver, remove the tip and form the 3 same sides, if you have a driver roller sharpener this will be easy. Now you have a stake to use on your roller by hand, take your time and be gentle !  If your stake is sharp you will be able to form the notches by pushing alone, roller will be soft.