Hi Andrew, there is no need to keep posting and re-posting the thread, it gets a bit unwieldy.
The SJR drawing I did shows a mast centreline about 7 inches forward of the original mast centreline, as best I can estimate from the drawing. I can't determine exactly where the hatch frame is.

How far forward can you shift the mast, while allowing for the necessary structure and allowing for the diameter of the mast?
I would not recommend raking the mast for a SJR. Too many things are compromised and you would need to design a SJR sail especially for it, and then it would be a very complicated sail to loft. Not worth it. There are a lot of good things about making this Amiina sail very simple and easy to set nicely, which you will find out for yourself, but if you alter the proportions, yard angle etc or rake the mast, you will lose all of them. It's a harmonised package, don't fool with it.
If it were my boat (and mine is very similar just a little smaller) I would say put a scaled version of the Amiina Mkll sail, on a vertical mast, minimum 6" forward of the original mast position, and you will be very happy with the result. You will be surprised how good that sail is. Make sure your mast head will be tall enough to allow for it to be higher than the peak of the yard (more is better) and with enough room below the boom for spanned parrel downhauls between the boom and the deck (say, 60% of panel width of clearance). No problem with a hybrid mast or a bottom tube extension, just don't skimp on height. I am sure Slieve won't mind if you want to contact me off line I can give you the dimensions of the Amiina sail, and you can discuss it further with Slieve when he is available. I will only do that on the basis that you do not alter the proportions of that sail without Slieve's approval - if you do, you're on your own. I think that sail has 33% mast balance - it works, anyway. Do not increase the balance from what it is in the drawing. 35% is taking a risk in my opinion.
To answer your other question (how to measure mast rake) I would suggest get the boat level on the trailer (can you assume the bunks are level? Put a spirit level on a bunk?) Then hoist a line with a plumb-bob and measure the offset. Some secondary school trigonometry will then allow the angle to be calculated. But you won't need to do that with SJR.
I wouldn't normally go this far, as I am not an expert, and I don't have a wide enough range of experience, but the Amiina Mkll SJR has been an unqualified success on my boat. In my opinion your rather similar boat, together with its hatch placement etc., just screams out for the same rig, with the mast as far forward as you can cram it, as the simplest and the BEST rig you can put on this boat. If the mast is a couple of inches further aft than ideal then I doubt you will notice the difference on the helm. (You will probably find this boat developes quite a bit of weather helm when over-pressed anyway, so the worse case scenario: a little more weather helm. Probably time to reef by then, anyway. Partly for this reason I would recommend that you do not go overboard with sail area - keep the sail area about the same as the original bermudan area). It's a powerful sail, and the usual argument "junk rig easy to reef so make a bigger sail" needs caution. Slieve will confirm this advice.
I will add that when heavily reefed my Amiina sail does not seem to perform so well, so there is another reason for keeping the working sail area within reasonable bounds. If the idea of maximum sail area for light weather and heavy reefing for medium weather appeals, there is reason for me to believe that the Johanna sail might be the better choice for you, as according to Arne it performs quite well with just the top fan triangle. If so, you might want to consider the bottom paragraph here.
With the above cautions I would say "Go for it". It looks to me like a great little boat, which should make the best of the SJR potential.
(PS Arne's suggeston of a mast placed forward of the hatch, with aft rake, is also worth consideration if you like to have a Johanna sail - plenty of good reasons for that. When there is no wind, the sail will tend to swing, under gravity, to the centre, rather than to swing and lie outwards. So what, if there is no wind? The other consideration is clearance of the boom tip when the sail is payed out, in a big seaway - the aft raking mast gives less boom tip clearance. But the Johanna sail is designed to have plenty of rise to the boom. And you won't be sailing in the ocean. For your little boat and the sheltered waters you will sail in, I doubt if there is any reason not to have an aft raking mast if it happens to meet your other requirements. (Personally I would prefer to keep the weight of the mast as far out of the forward part of the boat as possible, that's the only reason that would make me hesitate). Aft raking mast is a bit unusual for a junk, but so what? So there you are, thanks to Arne's lateral thinking, there is a good second option for you to consider. First, I would want to make a working drawing though, and further consultation with Arne. It's not certain to me that it will work clear of the hatch.)