I’m curious about this too… even thought about buying a Subharmonicon and looking inside to find out. Your plan seems the obvious way to approach it but step 3 is tricky, because it is not easy to keep the final sawtooth amplitude fixed: as the frequency goes up the cap has less time to charge so unless you adjust the gain somewhere, a double frequency wave would have half the amplitude.
Not that it’s impossible to add some gain compensation that depends on frequency, but it’s starting to get pretty tricky. (Edit: thinking about it a bit maybe that could be done with a current mirror from the main oscillator control current. Hmmm.)
When I was first thinking about this I came up with some other approaches, like using a counter to generate a staircase wave and then adding an attenuated version of the sawtooth on top, to “join the points” from the tip of one stair step to the next, but that just seemed ridiculous.