After some considerable length of time, I now remembered to post the resolution of the matter of the mysterious magic oil pressure in case it may be useful to anyone.
Experts at Rotax and at Lockwood (I saw both at Airventure Oshkosh) agreed that the oil pressure sensor had gone bad, as so many here had guessed. The oil pressure sender (sensor) was directly screwed into a port (fancy word for threaded hole) on the starboard side of the oil pump housing. The folks at Lockwood (specifically Tisha Lockwood) told me that FD had issued a blanket LOA back in 2008 allowing CTLS owners to "remote mount" the oil pressure sender. The LOA calls for the sender to be mounted on the firewall, beneath the battery box and air filter box, and next to the fuel pressure sender.
The advantage to doing so is immediately obvious: it keeps the sender away from the heat and vibration of being mounted on the side of the motor. The disadvantage is the cost of performing the LOA because it involves running both oil hose and fire sleeve, along with the required special clamps and fire sleeve sealing goop, and re-routing the wires. But it's a one-time thing, so I decided to do it. Lockwood supplies a kit with all the parts you need, including the hose and sleeve, the clamps, the nipples for each end, and so on. The kit includes a copy of the LOA. I had to order separately the fire sleeve sealing goop for the ends of the fire sleeve, and a supply of extra clamps after hitting the learning curve of doing twice-around "band-it" clamps. Plus the special tool for the twice-around "band-it" clamps and the special tool for the "Oetiker" clamps. Now I've got tools that will wait a long time before being used again, but the learning is worth it.
There's a tricky bit of wire-tying with a standoff to get the oil hose to route properly through the "U" bend in the number 1 cylinder exhaust pipe, and sit in the middle without touching the pipe. Hence the fire sleeve. But other than that (and the learning curve), it seemed pretty straightforward, although it took some time and patience to get it to look nice and tight.
The only part of the job that gave me any concern was getting the old sender off the oil pump housing without damaging the housing (or anything else). That sender was probably original, dating from 2008, and remember it was subject to heat and vibration on the side of the oil pump housing for almost 400 hours of operation. Getting a wrench in there is tight because of the fiberglass airscoop in front of that location. Tom Baker mentioned a specially ground wrench. I purchased a set of skinny wrenches to do the job. The wrench did get in there because it's skinny, yes, but also it is weak because it's skinny. The wrench actually started to bend, but luckily the sender decided to give up the struggle and began to turn, and then things looked much better after that.
For anyone facing this in the future: 1. Get the complete kit and the LOA (FD #080404) from Lockwood. 2. Get extra twice-around "band-it" clamps and the special tool for those unless you have those things already. Larger diameter "band-it" clamps make the going easier. 3. The "Adel" clamp in the kit is real tight, which is good, but I also got one the next size up (1/8 inch larger diameter). While still plenty tight, it made the installation much easier. 4. Oetiker clamps are supplied for the barbed nipples for the oil hose, so have that tool also.