Only the inspector can tell you the answer 100%, unfortunately. Theoretically, as long as you're only replacing the SEC + panel, and just reconnecting the existing subpanel feeders as they were prior, the inspector cannot make you bring that feed up to current code. Just like he can't make you install hardwired smokes in each BR/hallway/floor, GFCI all bathroom/kitchen/outdoor/garage receptacles, upgrade all 3-wire range/dryer branch circuits to 4-wire, etc. But you may get a cranky inspector and he's likely going to win any arguments.
That said, I'd prefer to pull the 4th conductor and bring that up to code as well. If you do that, you'll also likely need to install a couple new ground rods for the subpanel. Make sure you pull permits for the work, and let your homeowners know the work was done once completed and signed off - will reduce your premium.
Would also suggest installing a whole house surge suppressor while you're replacing the main panel. Eaton CHSPT2ULTRA work well. Supply houses usually sell them cheaper than retail outlets.