It seems like using two channels or a scene controller on start phase would work for one block. Or you could use two momentary control/other switches as bypass sources, with one also changing LFO start phase. The bypass modifier might need slight damping so the new start phase can take effect before the block engages.
BTW there's no real magic to the alternating sweep direction on the original. The LFO is just running around .425 Hz, roughly 3 cycles every 4 bars. Later in the song the alignment is a bit different and you can hear it reach min/max IIRC. A less repetitive version could use a bypass switch to get the right alignment some time before you actually need the flanger, plus a switch as 0-50% mix modifier.