Observations from the data:

• First row of data appears to be a statistical outlier and can be rejected: All readings are lower than the profile for subsequent rows, probably due to start-up behavior.
• The last band is not saturating, it varies from 1002 to 1022, so it is capturing valid data
• Bands 4, 5 and 6 are nearly saturated, but are not "stuck to the rail", they do show some variation, so they aren't in a failed, stuck-to-Vcc state

This indicates that the signal generated by the preamplified electret microphone is very high at the frequency bands represented by output 4, 5 and 6. This can be verified using a reverse biased avalanche diode or a zener diode (V_zener > 7 Volts) as a white noise source replacing the electret microphone in the BoB's preamplifier, and checking the output using an oscilloscope in spectrum analysis mode.

This is not surprising at all, as electret microphones are not very linear, and the preamplifier in the break-out board does not have any equalization built in, as per SparkFun's schematic. Hence net result is expected to be a rather non-linear output profile.

The MSGEQ7 frequency response, on the other hand, is pretty linear across bands:

If the problem just involves getting the signal levels for all bands into usable range, a simple resistor voltage divider to attenuate the incoming signal to perhaps 50 or 75% should do the trick, across all bands. 50% will leave plenty of headroom to capture signal spikes on any band, as will invariably occur with normal audio.

^{simulate this circuit – Schematic created using CircuitLab}

If it is also necessary to equalize the signal such that the mid-range and tweeter bands are attenuated while the relatively weak woofer bands are not affected, then a circuit such as a bandstop filter can be used, with a very low Q, and the pole frequency between the band-5 (~ 2200 Hz) and band-6 (~ 6000 Hz) frequencies. In effect, this will be more of a "band attenuate" filter, for the problematic bands, with some effect on the lower and higher bands as well:

_(source)

Such a filter is non-trivial, and perhaps deserves a separate question if required.

Personally I would go (and have in the the past gone) the FFT route. However I wouldn't use an 8-bit AVR, I'd use something somewhat faster.

My personal weapon of choice is the PIC32, which has more than enough power to do a good job of the FFT, but there are other good choices besides that - such as one of Atmel's 32-bit ARM chips (SAM3X for instance, like in the Arduino Due), or many of the other ARM chips that there are out there.

The trick with FFT is you need a good fast sample rate and enough memory to store a complex sample buffer. Say you want 1024 samples (which would give you 512 FFT buckets to play with), at 16 bits per sample, plus 16 extra bits for the complex FFT component, you're talking a minimum of 4KB of RAM just for the sample storage. Also, if you want it to be smooth then you want to be using DMA to read the samples in to one buffer while running FFT on another buffer, so a ping-pong double-buffer would increase that to 8KB.

With FFT you get half the number of buckets, or frequency ranges, as you have samples. The frequency range is also half the sample rate. So if you have 1024 samples recorded at 48KHz, that gives you a frequency range of 0-24KHz, with (24000/512=) 46.875Hz per bucket.

Reducing to 128 samples would give you 64 buckets, each at 375Hz per bucket.