Ethernet cable with largest conductor diameter
It seems a higher category (CAT5, CAT6, CAT7) does not necessarily
  mean the conductors used are larger in diameter as well.


Speed is the most important issue here and making the conductors thinner means that the capacitance between conductors is smaller and hence there is a greater likelihood that data can be transmitted at higher speeds. This runs counter to your requirement for bigger conductors and probably takes precedence.


  Any hints on how to identify the ethernet cables with largest
  conductor diameter / least resistance?


My 1st advice is to send power at a higher voltage level so that the \$I^2 R\$ losses are smaller AND, use an appropriate buck converter at the receiving end of the power. My 2nd advice is to use multiple parallel cores for transmission of power thus \$I^2 R\$ losses are reduced. But, be careful about using data lines with a noisy switching converter.