Andrew wrote: ...I've done some preliminary work on the CLR and CE which is well out of my comfort zone.
You really don’t need to do that. If your boat handles satisfactorily with the existing rig, then find or make a reasonably accurate drawing of the existing Bermudan sailplan and determine its CE. Then superimpose a scale drawing of your chosen junk sail so that its CE coincides (on the same vertical line) with your existing Bermudan CE. (The CE of the junk sail will normally be on a vertical line which passes through the midpoint of the boom, near enough).
Now that the two sail plans coincide in terms of CE you can expect the handling (lee helm/weather helm) of the junk rig to be about the same as the bermudan rig. Experience has shown this to be the case with cambered junk rigs.
Now, this will show you where the mast has to go, depending on the type of junk sail you have chosen.
You will have discovered that different junk sail planforms have different amounts of “balance” – the percentage of sail area ahead of the mast.
This is where you can experiment, on paper, until you find a planform which puts the mast where you want it to go. As David pointed out, the extremes are the Weaverbird wingsail type with ultra low balance, which puts the mast as far forward as you could go, - the SJR with its ultra high balance, which puts the mast about as far aft as it can go - and a range of “normal” HM and Johanna-type junk sails which can be found which will put that mast anywhere between these two extremes.
Summary: determine the existing CE. Find a junk sail planform, scale to the same sail area and fit it to the existing CE. Choose the type of junk planform which matches both the existing CE, and the desired mast position.
I hope this (not to scale, and slightly exaggerated) drawing explains how junk rigs of different balance give you a range of possible mast positions, with the CE matching the original bermudan CE.
This is not rocket science, it is merely geometry.

… use the existing mast and halyards to put the new mast in position, then de-rig the old mast and use the new mast to unstep it. That has been done before, successfully, and it is a very good idea.