My winning strategy:
1. Ignore units in the south
2. Directly approach the enemy base with my tiny number of units, somehow capture and sell construction yard & barracks
3. Only then capture the abandoned base
4. Directly build a helipad and use helicopter to keep the base alive long enough to build it out slowly
Step 2 took an insane amount of tries. Capturing the yard unlocked the heli pad for me (which I think is an oversight, it should not be available in the solo campaign).