I have not done this on a Hunter/quadrant, but I did change the headstock on a tiller-steered 33ft IOR 3/4 Ton racer while still in the water. Same deal, the headstock was the only thing holding the rudder in the boat, so removing it to replace required holding the rudder into the boat.
I went over it in mind many times, and since I had removed the rudder a couple times on-land, I felt I could do it.
I put a 3" ratchet strap around the bottom, water was clear enough I could check it for center from both sides. There was enough play in the rudder that I was able to confirm I had the full load of the rudder without the strap sliding forward/aft as I tightened the strap. I did an extra standard ratchet strap as a back-up.
Did my replacement... actually, added a split collar from McMaster below the new headstock, so in the future I would have a "safety".
Since I had removed by headstock several times on land, I knew how easy it would come off. That is one complication for your situation... I don't know how much you'll have to pry/twist/torque/pull... have to make sure you don't turn the rudder and cause the straps to slip. But I think it should be possible... maybe a chisel to open the quadrant or have a pulley arrangement figured out.
You could jump in the water and tie some lines around the front and back of the rudder, from one side of the strap to the other... I didn't, but would minimize concerns it could slide off.
Another idea, could you drill/tap a hole into the top of the rudder-post, then thread-in a heavy-duty eyebolt, use the main halyard thru a snatch block at the end of the boom to hold the rudder up. You'd have to slid the quadrant up the halyard, use a temporary pin thru the rudder to lower onto, unhook the the halyard, swap quadrants, and then take-up in the halyard again and install the new quadrant. Just a thought.