What is the method to measure EIRP (effective isotropic radiated
power) of a transmitting device?  What are the typical units of
measure for EIRP (dBm, dBW, etc.)?  Can EIRP be determined from a
field strength measurement (dBuV/m) if you do not know the antenna
gain (dBi) of the device under test?

The easiest way to measure EIRP it to measure the radiation pattern
of the antenna.  Finding the (say) 3 dB beamwidth of the antenna
tells you the antenna's gain factor (dBi) relative to an istropic
radiation pattern; assuming you know the power applied to the
antenna, you can calculate IERP.

Choice of units for EIRP depends on the rest of your calculation;
either dBm or dBW could be used - it's mostly a matter of convenience.
Or do it in (watts, milliwatts, microwatts, kilowatts, etc), and
multiply instead of add.

It would seem that a field strength meter would suffice for determining
the antenna beamwidth, assuming you're in the far field of the antenna
and have a suitably-sized test range or an otherwise anechoic environment.