How can RS-232 DTR / RTS pins be used for power?
I am currently using the DTR lead for two designs.  Like all RS232 leads, the voltage varies from a minus voltage (unasserted) to a positive voltage (asserted).  The voltage range is anywhere from ±3v to ±12v.  Frankly, I have never seen a voltage amplitude lower than ±5, and that was from a PC COMM port.

The spec for your device says to power the RS-232 interface using +12v on the DTR (asserted), and -12v on the RTS lead (unasserted), for a total of 24 volts.  It says it needs 20 ma.  This may or may not be possible depending on the RS-232 interface at the other end.

When I did this, there appeared to be a resistor in series with the line at the other end, which acted as a current limiter.  I "harvested" between 5 and 10 mA from the DTR line.  Even that caused the voltage to drop a bit.

I suggest testing this out first, by measuring the voltage on the DTR and RTS leads with a multimeter.  If they're not -12 volts, you can forgot supplying power this way.

If it is -12v, then put a 1210 Ω resistor across the DTR and RTS leads, assert the DTR line and measure the correspond leads to see if the voltage holds up (+12v).  You can assert the DTR line either in software (attribute of the COMM function in the Windows API), or for this test, use a program like RealTerm, which has the ability to control the various handshake leads under the Pins tab in its user interface.

I wouldn't worry about the handshaking; if they are telling you to power their interface this way, then it must not be an issue.