It still sounds to me as though there are two separate issues.
Whatever the malfunction of the strobes, the alternator should be able to keep the battery current positive after engine start.
There is no external strobe controller. The strobes are synchronized by a shared "sync" wire. The flash pattern is more complex than seen by the eye as each visible flash is made up of many shorter flashes. This scope trace shows the visible flash sequence. (Stobes as fitted to FX-3 serial 100.)