Most antifungal cleaners have got bleach in them, but I think that there are some that use only biological agents, I would think that your best bet would be to ask a pharmacist (Boots etc) as these might also have an adverse effect on your teak.
TôMö
It appears you have two problems. First the existing mildew, which would be best removed with just a damp cloth and no use of chemicals. Second, to prevent a recurrance, use a dehumidifier, since removing the dampness in the air will prevent the mildew forming. A few glass house heaters would help as well.
...migrates across the surface from the backs of the panels. Unless you can clean these areas too the problem will come back.
to get a reaslly depressinmg (but realistic) view of your problem view it at night by a torch shone across the surface of the teak.
On the positive side I found the spray made by "Starbrite" and marketed in chandlers under the name "Mildew Stain Remover" (look for a red & yellow label) very effective AND it worked well on cloth & vinyl too
I have an aging W30(1969) the only area that I have mildew, and very little of it is in the head, this is easily removed with a cloth and warm soapy water. To eliminate all mildew I removed all of the origonal lining, that was to deckhead, hull sides and all vertical and horizontal surfaces.
I completely relined with corded carpet and have not had a mildew problem anywhere! just a thought.
I would wipe down not with bleach which is quite corrosive but with spirit on a cloth. Then resurface the wood with some protective - beeswax or a proprietary wood sealer - not paint.
Spraying to reduce the spores in the boat, particularly in auturn before winter storage, is vital to preserve all fibres and timber finishes in the vessel. Keeping the boat aired all winter long, under some cover with plenty of ventilation is key to reducing the conditions for mould spores to multiply. You are right not to let the situation go as permanent damage and unsightly marks will be left in time.