The tendency to fire, then lurch and not make it to the next firing position is interesting, and it is with this characteristic in mind that I make the following suggestions. However, now that you have a consistent symptom to describe, the first thing to do is ring Bert Flood. I have found them to be very helpful - they may know straight off what it is.
In reading your post I get the impression this problem started one morning when you tried to start the engine, versus a problem that just got worse over time. When my engine is very cold, or has a low battery, it eventually fails to wind over. It has never thrown a tantrum as you describe. Also, the symptom is consistent, and now occurs every time. I am also assuming you have not just modified, or changed something between when it was working, and now when it is not.
If this is so, I would be looking for something that is actually broken, failed, or disconnected, rather than something that is loose, or out of adjustment, or marginal in its operation. In this case continuing to try to start it, or getting a bigger battery is not the answer: you might be doing damage.
Since it is firing, I would rule out fuel, choke, coil, spark plugs, carburetor, etc for the moment. No doubt you would have turned it over all compressions by hand (with the ignition off) in your pre-flight to see that there is nothing obviously untoward.
I can think of two reasons the motor could be failing to turn over after firing - the ignition system is firing at full advance, or there is a failure of the clutch to impart enough torque to the propeller for the propeller to carry the engine through to the next firing.
The Rotax engine has a dual ignition system, and I think the modules incorporate the automatic advance. There do not seem to be any parts in common between the two systems, although I don’t know how or where the trigger sensors are mounted. (if this mounting has become loose and moved, you would have had an exciting last flight). It is unlikely both modules would have failed in the same way at the same time. To confirm that he failure to start is not caused by one of the ignition modules firing at full advance, try them one at a time. If you have a select key and a start button, this is easy. On my engine, this is not the case. Both modules are supplied with 12 volts by their own red supply wire right at the module, and it is easy to unplug these one at a time there.
In the Rotax Line Maintenance manual (which I downloaded from Rotax site on the internet) there is a procedure for checking the setting of the overload clutch, that does not involve disassembly of the gearbox. It does, however, require some work, so get your overalls on.
If that fails, rip some more pages out of ERSA, and call Adrian.
Jack. :):)