If you are concerned about checking for a wide variety of potential spurious emissions, you will want to look into the many software packages for using the RTL-SDR as a spectrum analyzer. The RTL-SDR.com blog has a number of articles on this, e.g. http://www.rtl-sdr.com/rtl-sdr-spectrum-analyzer/.
What these do is programmatically tune across a whole wide range of frequencies and sample the spectrum at each "stop" along the way. So instead of seeing only distortion products in the ~2MHz spectrum around one particular center frequency, you can get a single composite view from the whole tunable range with one of the "spectrum analyzer" helpers.
Of course, you can manually accomplish something similar simply by tuning to various windows where you might expect to see spurious signal, e.g. 2x the transmitter frequency and further harmonics, or at various combinations of signal/IF/LO interactions. But scanning through an entire wider range while continuously transmitting might help you catch something you wouldn't have thought to check.
Note also that the signal on your handheld transmitter's antenna port will be waaaay to strong to inject directly into pretty much any receiver, including your SDR. So if you want a direct cable connection, make sure to use an appropriate amount of attenuation. Even picking up the signal via antenna at the same workbench could easily overload the SDR's receiver circuitry and cause a variety of additional signal distortions to show up, which aren't actually emitted by the transmitter itself. Make sure your receiver is not overloaded by your (or any other nearby transmitter!) before considering its output valid.