I think you could do it.

There are two reasons why it is normally not possible to put a junk rig onto an existing bermudan (or gunter) mast.

The first is that the mast itself will probably not be suitable. A junk rig really needs a free-standing mast, not necessarily as tall as a bermudan, but larger in diameter, capable of standing without the support of rigging, and generally mounted on the keel, supported by suitable deck structure, partners etc. A mast designed for standing rigging is usually not strong enough without it. That is an engineering problem, which you would need to solve by finding a suitable mast or tube, and ensuring strong enough partners at deck level. For example, you might be able to make a hybrid mast from a suitable piece of aluminium tube for the lower part, and using your existing short gunter mast, for the upper part. There is plenty of information available on this site, on the subject of hybrid masts. There are also other solutions for a home-made free standing mast.

The second problem is one of geometry. The replacement sail plan must have a geometric centre (CE) which approximately matches the CE of your existing yawl rig. (Otherwise, you will have too much weather helm or too much lee helm). It usually results in having to shift the mast position forward - it is rare to find an existing mast position which suits.

However, you have some trump cards to play.

Let us leave aside the mizzen sail for the moment and focus on the gunter sloop rig which would be replaced by the junk rig. You would need to find a junk sail design which would suit that mast position, with a CE matching the CE of your existing mainsail and staysail. I made a rough sketch based on a sailplan I found on the internet, but you would need to make a better drawing with a bit more accuracy. Most junk rigs have a CE through which a vertical line will run approximately through the half way point of the boom. You fit that CE to coincide with the CE of your sloop sail plan – and then observe where the mast lies in relation to the sail.

Now, here is the tricky bit – the mast will divide the junk sail into two parts. The ratio of sail area forward of the mast centreline, with the area of sail aft of the centreline, is a ratio called the “balance”. Most junk sails have a balance somewhere in the range of 10% to 25% (with some extreme examples outside of this range) and it is this “balance” which will determine what type of junk sail (if any) will fit this mast position. A rough example (as shown here) suggests to me that the “balance” of a junk sail to suit your yawl mainmast position, will need to be at the extreme end of the range – 35% or a little more. Or, to put it another way, most junk sails will end up being too far aft, to suit the geometry of your yawl sail plan.

Your case is special: because of the mizzen sail, your main mast is further forward than normal.

One solution is to dispense with the mizzen altogether and design an entirely new single-mast junk rig based on your existing mainmast position. It is my guess that one of Arne’s “Johanna” sail plans would fit perfectly. An accurate drawing would soon prove whether this is or is not the case.

Another solution, if you want to still have the yawl rig, is to find a junk sail which matches just the gunter (main) part of your rig. That will likely require a sail which has more “balance” ie more sail area forward of the mast, than a conventional Johanna type. It is possible that a “split junk rig” (SJR) would fit. I have superimposed a scaled down version of Slieve’s Amiina SJR sail which has a “balance” of 33 - 35% (the maximum possible). I think it is still likely to be slightly too far aft, but I think it would work if you were to reduce the mizzen sail area a little – or increase the SJR a little. Also, with a dinghy, you do have a little bit of flexibility with the way you deploy your centreboard, and how you trim the boat.

The Johanna planform and the Amiina planform are both well-tried and proven junk sail types, with documentation available, which people have made themselves. (Junk sails are not necessarily well-understood by ordinary sailmakers - a warning here, do not entrust this job to a sailmaker unless they are experienced at making a junk sail. Most are not.) 

Those are the broad ideas you need to consider. If you were to respond to this by being a little more explicit about what you want and what you feel capable of, together with a decent drawing of your existing sail plan, preferably with your CE of your existing yawl rig marked on it, my guess is that in your special case a junk rig can be found which matches that existing mainmast position. I happen to rather like the Amiina SJR, though your most straight-forward solution, in my humble opinion, would be one of Arne’s Johanna sails, probably without the mizzen.

PS a rather delightful junk yawl rig was designed some years ago by an experienced sailmaker in New Zealand (Paul Thompson) using a somewhat different "fanned" junk sail planform. The masts are at each end of the boat. It was very successful, and here is a short video of it sailing home from a junk rig meet - we call these meetings "junkets".Freebie Sails Home

I am sure you will find a solution.