How to improve filter quality at low fs
The impulse invariance method as well as the bilinear transform will give you relatively large errors close the Nyquist and there is nothing you can do about it. One thing you could do is use a filter design method that approximates the given analog response. I've tried this for the RIAA de-emphasis fitler with an (unpublished) algorithm I came up with some time ago, and the figure below shows the result:



The digital filter is a second-order IIR filter, the sampling frequency is $f_s=44100\text{ Hz}$, and the error is much less than 1dB over the whole frequency range up to Nyquist. The price you pay for it is 20 iterations of an approximation method, which solves a $5\times 5$ system of linear equations in each iteration.

EDIT:

If you add one more pole and one more zero (i.e. your discrete-time filter has 3 poles and 3 zeros), then the error goes down to less than 0.1 dB over the whole frequency range up to Nyquist: