How can I filter dissolved PVA from water
Rig up a column that you can fill with granular activated carbon (GAC). Throw batting on either end to prevent carbon fines from entering your dissolution chamber. Replace the carbon when you reckon it is no longer working. If you can afford the heat, throw a cooling coil in front of the filter.

Another method is indicated in this patent: US4078129. They add 0.25g sodium borate and 0.6g sodium sulfate per liter to precipitate between 0.03 and 3% PVA.
This may not be a good solution if you are prone to overdosing, as it would say, goop your soup, when you go to do the next batch. 

A third method seems to be already popular at the large scale: ultrafiltration. There are many papers discussing its usage for waste water discharge applications. Use the smallest pore UF you can find (50kDa is what I've seen in the literature). The pressures required to operate it are pretty low. Be sure the temperature of the solution is &lt;50 degrees C prior to entering the UF (almost all membrane applications fail >50 degrees C). Run it outside-in. Don't dead-head it. You will generate a small concentrate stream. Perhaps try method 2 on that to reduce waste generation.