Suggestions for 200-600 nm range transmitting polymers/materials
No plastic of any practically useful thickness$^\dagger$ has $80\%$ transparency down to $200~\mathrm{nm}$. Period.

Below is a UV transmission chart for various materials from a marketing brochure published by BrandTech, a manufacturer of disposable UV/Vis cuvettes (click image to enlarge):



The chart only goes down to $210~\mathrm{nm}$; extrapolating the trend, not even quartz is $80\%$ transparent at $200~\mathrm{nm}$, but it's probably a decent choice for your purposes as long as it sufficiently meets your other criteria.

Beyond that, fluorite $\left(\ce{CaF2}\right)$ is the only specific material I know of with better UV transparency; it appears to have better than $90\%$ transparency down to around $200~\mathrm{nm}$ (figure is from ThorLabs online catalog):



A chart at that ThorLabs link indicates two additional candidate materials: $\ce{MgF2}$ and sapphire. As with quartz, the choice among these will boil down to how well they meet your other criteria.



$^\dagger$ As a caveat to the above: If a very thin plastic film ($\leq 100~\mu \mathrm{m}$ or so) is acceptable, a specialty material might give sufficient transmittance. You would have to convince the manufacturer of such a material to make it for you, however, which would probably not be trivial, or cheap.