This is the sort of amplifier I would consider using between your sig-gen and the 10 turn CT.
http://www.ebay.com.au/itm/191952191816?_trksid=p2055119.m1438.l2649&ssPageName=STRK:MEBIDX:ITNo. That would produce 7kV. Your measurement was wrong.Yes, but that is more than you will need.
You are using a current transformer backwards to inject current into a common mode bundle of wires. The CT turns ratio is then 10 : 1.
Since power in = power out, the output voltage will be 1/10 of the input voltage. The current induced in the output will be 10 times the input current.
You measured 5.3mV output voltage, so you probably had 53mV input voltage.
Your signal generator was unable to do better than that when driving the 10 turn CT.
We do not know the circuit or circuit resistance you used to measure the 5.3mV.
We do not know the output current capabilities of your signal generator.
You do not have to use the 10 turn CT secondary.
You could wind more secondary turns onto the CT core to change the ratio.
I would try 50 or 100 turns, driven directly by the sig-gen.