OK, I finally got the desired result. 95% of the time went into restarting Fusion because it almost always crashes when inserting an svg file into a sketch.
Here's what worked: Generate puzzle cut pattern using online tool http://www.wolfiesden.com/Laser/PuzzleCreator.asp
Edit that pattern in Inkscape .91
First delete all the vertical cuts and save the horizontals, then reload and delete all the horizontals saving the verticals.
Use Inkscape tools to extend the cuts past the edge of the desired area and tie them together with curves (not necessary, but maybe handy--fewer tiny profiles to select). Draw with stroke set to the kerf width desired. I used 1 point, =1/72 inch. Take great care to make sure the lines don't have tiny disconnects in them. I found half a dozen 0.026mm gaps, which caused great trouble. It takes a lot of zooming in Inkscape to be able to find and see such tiny gaps. There must be a tool that closes tiny gaps, but I'm not an Inkscape user normally.
I chose rounded end caps for the lines.
Then convert the stroke to paths, which gives a closed profile that I can extrude later, and save the file(s).
In Fusion, create an offset plane to sketch on, new sketch, and insert the horizontal cuts. Save. (It may take 10 tries to insert an svg without a crash. Starting from a brand new sketch may help.) Select the tiny thin long profile and extrude it some useful distance as a new body. Hide the sketch and extrusion. Make another new sketch (I put it on a different but parallel plane to try to keep the complexity of one from somehow affecting the next.) Import the svg for the vertical cuts. Show the previous extrusion. Extrude the vertical cutter profile similarly, using join option. Now we have a cutter.
Then make visible the items to be cut--it would be better to combine them first, as you can only cut one body at a time. (I tried splitting body, but that always failed to compute, while combine/cut worked.) Use combine, with the cutter tool and cut.
Done.
The hard part was finding a pathway that Fusion could handle, and most of the time was spent waiting for Fusion to reload after the svg-insert crashes. Some of the early crashes were doubtless caused by the complexity due to tiny gaps, but Fusion still crashes very reliably even after the curves are smooth and the gaps are gone. You just have to save often and keep trying. Usually the crash happens during the move-into-position phase of the insert svg, but sometimes it doesn't get that far. Painful.
Now I want to change the puzzle details and sizes etc etc, but it's so tedious I'm kind of inclined to live with what I got.
It's a real pity one can't edit such curves in the sketch. Constraints refuse to display, etc. Necessary to use some tool like Inkscape.