Paul, from what I understand fungii spores are omnipresent, what causes them to grow is moisture and warmth, IMO, treating the fungii without reducing the moisture levels will be a waste of time. If it's growing on the floor then chances are it's taking hold elsewhere.
Tin sheds are notoriuos for moisture buildup, I'd be installing a small reverse cycle AC and running it during moist periods to dehumidify the air. If that doesn't appeal then some form of dehumidifier to reduce the moisture content.
Remember the C14 at Mapleton....
As a last resort, you could always move to a drier climate, Tasmania perhaps...
