I would expect that gilding - classic manual gold plating using gold leaf - works.
(But as the frequency, amplitude and radius of flexing is not known, it's hard to tell)
It would be starting with the polished polymer surface.
That is coated with a special glue,
the gold leaf is just layed down on it,
and the gold surface is polished/burnished to create the mirror surface.
The difficult part may be finding the right glue substance, which depends on environment conditions. But a standard product should work.
So - why do I propose this ancient method?
I see the advantage in using an actual (microscopic) compact layer of sheet metal.
Gold leaf is just a thin gold foil.
But there is a special property in gold: it's very ductile. It's so ductile, that gold leaf can actually be produced roughly by taking a piece of gold and hammering it flat.
Now, I would think that the gold layer will not have much problems with the flexing polymer under it.
Another important property of gold here is the innertness of the metal, because we have the mirror layer directly exposed to air - not behind glass like in "consumer mirrors". With silver, that would be a problem
Gold makes a very good mirror for many wavelengths - you did not say which you are interested in. But if the yellow color tint of reflected visible light is a problem, one could try the same with other metals. No other metal has the level of ductility of gold, so they are possibly less suitable.
What makes me think that creating a sheet of metal first, and putting it on the surface in a second step is that any kind of metal plating where the metal is deposited on the surface will follow any roughness of the surface, which will cause irregular thickness and mechanical strength. On the other hand, the layer following the contours has a larger area, when looking closely, which shoud help with mechanical stretching.
The typical thikness of deposited metas ist thinner than a foil. It can be made as thick for sure, but this could make coating expensive in terms of processing time.