Thanks for the many replies. The division of opinion mirrors the uncertainty I began with, whether the spacing will be the same with a new freewheel and allow the shifters to work.
The Sheldon Brown cribsheet linked to above shows at least some Shimano and some SunTour stuff having the same spacing, but it’s not clear to me whether that covers this case.
Specifically, the cribsheet shows the SunTour “Ultra” freewheel with same spacing as Shimano HG. It also list SunTour Microdrive with variable spacing, but it looks like those were cassettes, not freewheels. But I can’t tell whether the cribsheet purports to be a complete list of all possibilities.
Since I am planning to replace this bike soon (but keep it as a backup/winter bike), I’m reluctant to put a lot of money into replacing the whole system.