but if they execute that END block when they exit...

They do. Even without the END block, the problem would still exists since the child would call $dbh's destructor which calls disconnect (after issuing a warning).

Sorry, I don't have any good solution short of connecting after the fork. Maybe you can create a pool of reusable children early on? Maybe you can exec within the child, even if only to relaunch Perl? Maybe you can add to a job list which monitored by cron job instead of forking?