If @Hags or anyone else wants a workaround until the fix is properly released, this worked for me (thanks to Ralgha for the suggestion):
Open the mission in the Mission Editor ([DCS install folder]\Mods\campaigns\The Museum Relic\MR campaign 1.miz)
Zoom in to Anapa airport and find the black circle around it (the airport's capture radius)
Add a Red faction ground unit roughly 100m outside the black circle (e.g. a Russian BTR-80)
Give the unit a waypoint roughly 100m inside the black circle
Start the mission (you can't save missions from campaigns, but you can play them directly from the editor)
Wait 30 seconds or so, and the ground unit will capture the airport, and wake up the ground crew
Complete the mission as normal (this doesn't count towards the campaign progress, but you can score it as a moral victory)
End the game, go into the Campaign screen, and restart the mission from there
A few seconds after mission starts, open your communications window, and select the "Other" -> "Auto-complete" option to insta-win the mission
End the game, and return to the campaign screen, now updated to show you've progressed to mission 2.