Why does a MP3 encoder use a fast Fourier transform before applying the psychoacoustic model?
I would suggest a more detailed explanation of mp3 codec.

FFT is applied on the time domain signal, so in fact it does not use the result from the MDCT.  The input to the psychoacoustic models is in the frequency domain, hence the FFT.  

There at least several reasons to do it.
MDCT with filterbanks operates on very short overlapping chunks, maximizing the compression&nbsp;&ndash; the FFT uses longer samples and has better spectral resolution. (It is hard to compare since MDCT operates as short-term transform; if this is of big importance to you I will have to make that comparison.)

You can think of filterbank MDCT in the same way as JPEG quantization (it is a very good analogy, since both use DCT) and FFT to detect DCT artifacts from compression. 
Then, the psychoacoustic model smooths the errors to fall under the "hearable" threshold, but in order to do that, the time domain samples (here PCM&nbsp;&ndash; Pulse Code Modulation is not enough, because sudden frequency shifts are heard as cracks) - so it uses the frequency domain to detect such discontinuities and then smooth it in time domain.

Two things are not explained in articles but are crucial. When PCM differences, are high the speaker has more distance to travel, so there is time delay and depending on speaker abilities it might just cause additional vibrations, which are quite distinct noises from speaker. 
The second part is between lines, the quantized version of signal is transformed back to compare it with original sound and check how much it deviates.

Based on the masking type of windows (based on comparison of FFT and inverted MDCT) is chosen to compensate better for audible deviations from the original.

Humans perceive frequency shifts better than amplitude changes, so the filter operates in the both domains at once, and the quantized signal is reversed and smoothing is done back in time domain.

Yes, the resolution of MDCT with filterbanks is not enough, but this is the part where a fair share of the compression happens, and then it is masked. But the psychoacoustic model has spectral resolution as given in the paper.

Yes, FFT is more accurate because it gets longer samples, so it has better resolution between bins.

Footnote
The (M)DCT is commonly implemented by performing FFT, so this has nothing to do with transform used. MDCT can be seen as a bit-modified Short-Term Fourier Transform with a specially chosen filter (the filterbanks resemble Mel scale for speech recognition).  

FFT is used longer, provides easier algorithms for pitch shifting and is easier to apply on sound. 
(M)DCT minimizes the number of components, meaning that we can cut more data from the result than from FFT.  

But in the case of sound those components are not stable, cutting always e.g. two bins out will give bigger disortion between consecutive frames than doing equivalent operation on FFT results. 
So, the connection between FFT and what we hear is bigger than (M)DCT and what we hear, but available compression is the other way around.