Up to 700mA.
The idea was to use thermally conductive tape to attach the boards to some kind of heat sink.
how about using jumper wires or 0 ohm resistors instead of the 2nd layer, then your pcb backside can be completely dedicated to thermal area... since the thermal pad is electrically neutral you can just do a big copper pour, and maybe specify no solder mask for that layer as well, to work better with your thermal tape. nice fat traces on a pcb look impressive but given the short distances you're dealing with, smaller traces would give you more room for jumpers. keep the fat pads for good mechanical stress handling of your mounting wires.