Our students in electroacoustics use this method in a lab on
loudspeakers. We use a microphone inside the box (Sennheiser MKE-2)
and software that can tilt the response by 12 dB/oct. It works great
(within the decibel compared to theoretical predictions of the
frequency response) down to the lower frequency limit of the
microphone and up to about half (or something) of the frequency of the
first standing wave inside the box. In theory, if you only could find
a microphone good enough, it would work down to DC. Works for closed
boxes and bass-reflex boxes.

The lab instructions are in swedish (sorry) but can be found at
http://www.speech.kth.se/~svante/elak/htlab.pdf
I think you can get an idea about it by looking at the pictures :-)

The software can be downloaded at
http://www.speech.kth.se/music/downloads/smptool/

I thought up my own method for eliminating the effect of the room on
bass measurement (although, no doubt it has been done before). The
main feature is to place the (pressure) microphone within the speaker
and to compensate by 12 dB/oct.

Normally I use a MLS and filter out higher frequencies, say above
100Hz, with a brick wall FIR filter using Coooledit. The 12 dB/oct
compensation is also done in a similar way, either to the input MLS
or to the output recording. This sort of pre-filtering can give a
large increase to the signal to noise of the meaasurement.

There must be some restrictions on the accuracy of this method but
the only one I can think off at the moment is that the wavelength must
be large compared to the internal dimensions of the speaker. The
meaasurement includes the effect of any port and of the enclosure
flexing.

There is probably an assumption that the air is compressed
adiabatically. The pressures involved are, of course, small if the mic
is not to be overloaded.

No, the pressures involved are HUGE, typically there is no problems
reaching 140 dB inside the box at low frequencies. So keep the levels
low in order to avoid overloading the microphone. (Or did I perhaps
misunderstand your statement?)