Hi Mike, welcome to Outdoorking.
It is not unusual for Honda GCV engines to break exhaust rocker arms, but there is always a reason, and it needs to be found.
When a rocker arm breaks there are two things to check for immediately.
The first is whether the valve is stuck. That is by far the most common reason for GCVs to break rocker arms, and it seems to be often associated with the engines being fitted to pressure washers. Water gets into a lot of the wrong places on pressure washer engines. A stuck valve is easily found and it will be obvious if that is the problem. Since this has been happening to you for a while and the engine has been running along the way, it seems unlikely that you have a fully and permanently stuck valve.
The second thing to check is that the valve will move as far as necessary. The valve lift is set by the camshaft and the rocker arm lever ratio, and on the GCV engines there is only one (usually plastic) cam lobe which operates both valves. Since you are not having any problems with the inlet rocker arm, chances are the problem is not with the camshaft. However, you still need to check whether the exhaust valve will freely open as far as necessary. I think the easiest way to do this, is to assemble the engine with (another) new rocker, not a cracked one, adjust the tappet clearance correctly, and rotate the crankshaft in the forward direction until the exhaust valve begins to open. Then, press down on the rocker directly above the valve, and note that the valve will open further. Rotate the engine further in small steps, checking at each step that the valve will still open further. The objective here is to see that the valve does not run out of available movement, even when the maximum lift position is reached. If it does run out of movement, when the engine runs the rocker arm will have to act as a flexible joint every time the valve opens, and it will soon fatigue and fail.
If you find there is insufficient valve movement to accommodate the full lift dictated by the camshaft, you know why your rockers are breaking and you just have to find what is causing the exhaust valve to have restricted travel. There are four obvious possible reasons. The first is that the valve guide has moved upward from its correct position. The second is that the valve has been adjusted with insufficient clearance, which has the effect of increasing the valve lift. The third is that the timing belt has jumped, and the head of the exhaust valve is hitting the piston, which limits its movement. The fourth is that there is incorrect assembly of the valve retainer, so that some part of the retainer system, or some additional component which should not be there, hits the valve guide and prevents full opening of the valve.
I know you have checked two of those four things (correct valve clearance, and correct belt timing), but if it is neither of the other two, I suggest you recheck. Then, if it is none of the four things, we are reduced to looking for unlikely problems such as a damaged or incorrectly machined cylinder head or rocker pivot, causing the rocker pivot geometry to vary when the engine is running.
Please report back when you have made these checks. If possible, include some pictures.
