There's a bit of a problem in how you define "laser weaponry".
To my mind, modern industrial laser hardware really isn't comparable to laser weaponry, because the peak powers are generally just too low. Unless each pulse of your laser is delivering a few tens of joules in no more than a couple of microseconds (and hence your laser is, at a minimum, developing peak powers in the megawatts) then what you've got isn't really very useful as a weapon, and will compare extremely poorly against any other kind of weapon, including railguns, coilguns, guided missiles, unguided rockets and plain old chemically-driven guns.
Once you've created a laser capable of this sort of performance, the thermal conductivity of the target is more or less irrelevant because there's insufficient time for the heat to be conducted away before the material has been evaporated. At that point what you need is an armour that is tough and refractory, and copper is neither. Conventional steel armour plate is much more practical in the face of that sort of weapon, because its melting point is much higher as is its heat of vapourisation and even its specific heat capacity and its a hell of a lot tougher to boot. That means that practical lasers will make shallower, narrower holes in steel armour than they would in copper but perhaps more importantly it protects against conventional weapons too.
If your enemies are using lasers that can be effectively defended against simply by wearing copper plate armour, then your enemies are poorly-equipped idiots and you should thoroughly teach them the folly of their ways using plain old bullets, bombs and shells. Bonus points if you can trick them into wearing copper armour to defend against lasers, which ends up being a logistical and economic nightmare and fairly ineffective against decent guns.