The return path for a trace in a single layer flex cable in free space is going to be the other traces. If this is a digital interconnect with high speed, you most likely have a layout like:
gnd - sig - gnd - sig - ...
gnd - sig - sig - gnd - ...
This is typical coplanar transmission lines and you can simulate that easily with a 2.5D simulator like Hyperlynx og SigXplorer (both expensive). I think you can get a good idea from the free TNT field solver as well. That will help you determine both what the impedance of your signal traces are and what amount of coupling you get between traces.
Now if you put this (very) close to other metal like your top layer Cu this may change. If and by how much depends on the geometries. Put Cu closer than the trace-to-trace distance in the flex and it will certainly do something.
Specifically it will reduce coupling (x-talk) like you mention and lower the trace impedance.
It it worth doing? Good question.
I would recommend you do some quick simulations to find out before you waste too much time and energy solving a problem that may not even be there to begin with :-)
Let me know if you want to provide more details, so I can give a more detailed answer?
As can be seen on the photos you added, the flex is a 2-layer and some of the traces actually do have a reference plane running on the opposite side of the cable.
Provided the thing is well engineered, you should not try to change impedance by putting other metal real close to the cable. Don't however assume that it is well engineered just because some (well known) company is selling it :-)