bacterial & fungal are easy to sort, put a bottle of bleach in the
header tank, leave it there for a bit, then draw enough water for it
to enter the cylinder. Flush out through all the taps later. At least
you'd eliminate one major possibility that way.

I agree its probably bacterial growth, with a firm hold on part of the
system walls - hence it dislodges under high flow.

Safer than bleach though, is water purification tablets (the type for
using in drinking water in dodgy parts of the world - buy them in
Boots) - chuck a whole packet of them in the header tank.