I had a 1250 for several years. I did have some similar symptoms before I finally traded it in, the coil was bad. It seemed to have great spark under normal diagnostic conditions, (hold the plug against ground and look for spark) but after ruling out fuel and compression, I tried a coil from TSC. It was a 12v coil and after I hooked it up it started right up. Mine would also run no problem until I shut it off and then not start. Problem was the repair didn't last long, about three mowings. (about 15 hours) I finally spent the dollars on a genuine part and that seemed to fix the problem. If I remember right, the coil mount broke about the same time and I ended up mounting the coil to the plastic barrier around the gas tank. I think that helped the coil lifes seemed that the vibration had a negative effect on how long the coil would last. I did run a ground from the frame to the coil mount, I think that also helped.
Good luck,
DTS