How to find an EAGLE routing variant which will be OK after final processing of polygons
Is there anything I can do to make EAGLE check whether they really are 100% routable, before I permanently select one?
  


No. 

What you can do is force connectivity between islands. Pick a routing variant you like, identify polygon islands, unroute (a simple undo would do), add a ground trace betwee pads to connect the islands and then reroute. 

Using an autorouter is a questionable issue. Many dislike it. I found it to be very useful in my hobby projects. First route manually the critical nets (power/ground, crystal, fast data, etc) and then let the auto router do the rest less critical nets (~60-70% of the nets in my designs).

Another trick is to play with the costs to encourage less and shorter traces on the polygon layer.