Backup Generator interlock question
You've got a very good grasp of the situation already. At a high level the options would be

add a main breaker to the existing panel

if one is available
if a compatible interlock kit is also available
if you make room in that already-full panel for a backfeed generator breaker


replace the existing panel with one equipped with a main breaker and more room
or, add another panel.

Personally I'd lean toward the latter. Choose a panel that has a main breaker, feed-through lugs, and supports a generator interlock. Insert it so that the existing meter-main panel feeds into the main breaker of this new panel, then power the existing subpanel via the feed-through lugs of the new panel. If a panel with feed-through lugs is too difficult to find then the existing subpanel could instead be fed through a branch circuit breaker in the new panel.
This arrangement buys you a bunch of empty slots for future circuits, the opportunity to put the panel in a more convenient place if desired, and minimizes change to what already exists.