Yeast wild or tame die just as easily in cleaning chemicals as most bacteria, they are not special super organisms, oxidation and reduction destroy their cell membranes just like any other cell. Including ours so always follow instructions when using chemicals and wear hand and eye protection.
Regardless of the material your FV is made from if your sanitation and cleaning are sufficient to prevent contamination normally then you should be absolutely fine. One added precaution I would take is to ensure all pipes and fittings are dismantled and placed for 15min in water at or above 90C.
If your fittings and flexible tubing are sterilised, and you wash down your FV, then sterilise with bleach or caustic, then rinse, then use a no-rinse sanitiser you will have no problems.
If cross contamination occurs, first re-examine your sanitation processes, then inspect your FV, fittings, seals, and tubes and potentially replace the old if scratched or pitted, as if Brett or Pedio can survive from one brew to the next so can lacto and Acetabacter and it is only a matter of time before you have spoiled beer.
I speak from experience, I had issues in the past and my sanitation was not good enough, I changed my process since that time I have had no issues, we are talking over 4 years now Wild and clean fermentations on the same kit, no cross contamination.
I use Caustic, Phosphoric and Peracetic Acid to clean all my vessels, be they stainless, glass or plastic. If you do not have access to these then Caustic, Bleach, a water rinse, StarSan works very well.
Don't use bleach on stainless unless it is 316L or it will damage it.
Strong caustic with prolonged contact time or heat will etch glass.
Never use above a 3% solution.
Always add chemicals to water not water to chemicals when diluting.
Always wear goggles and gloves when handling chemicals, and long sleeves and long trousers.