Very likely the frequency range of the DVM doesn't go to 300 kHz.
I would go for a (shielded) current probe and an oscilloscope. If you don't have an oscilloscope you could use a current probe with a two diode (voltage doubling) rectifier, but that requires serious design as you have low current level.
You can make a current probe yourself from a high permeability ferrite with sufficient secondary windings that go to a 50 Ohms input of an oscilloscope. Of course you can also load the current transformer with a resistor and use a high impedance probe. You can't make the current ratio too small, otherwise your signal may dissapear in the noise.
Don't take for granted expensive looking current probes. When the impedance level at the measurement position is relatively high, the current probe may respond to voltage also. That is why I prefer shielded current probes (own construction) for probing high impedance circuits. If you have access to the return (ground) wire, you may put the current probe around the return wire reducing respons due to voltage.