OK, here’s how it turned out…
As I stated in the question, the trick was to try to get the tile up without pulling the linoleum up. The newer tile is less than ten years old, no danger of asbestos. Some of the tiles have broken or chipped. (But mostly because my wife said to!) The older linoleum was installed when the house was built, circa 1978, so I don’t want to start ripping it up because of the danger of asbestos.
At first, I started working a paint scraper betwen the tile and linoleum to loosen the adhesive and pry the tiles up. This method, while effective, was physically difficult, and very time-consuming.
After a couple of days (yeah, days!) of this I had made very little progress, and was aching all over. Then I tried a wide chisel. That was a little better but it tended to bite into the linoleum, and I didn’t want to risk exposure to the asbestos.
I finally used a variation on @gurnblansten‘s idea of a propane torch. Only I used an electric heat gun, the kind used for shrink-wrapping. It loosened the adhesive to the point that I could pri each tile up, almost intact (not that it mattered about the condition of the tile; it was easier to handle and dispose of). Ass I removed a secdtion of the old tile, I immediately put the new tile down using the adhesileft on the floor and on the back of the new tile.
Here’s how it came out!