Optimal tradeoff between ADC bit depth and sampling rate
Averaging sets of 24-bit samples is essentially applying a filter with a rectangular impulse response, which leads to a frequency response of a sinc function. The peaks in the tails of the sinc function will alias some of the noise down into your band of interest.

Nevertheless, simple averaging could work well. For example, averaging groups of eight samples at the transmitter reduces the Gaussian noise to

$$ {19 \textrm{ LSB rms} \over \sqrt{8}} = 6.7 \textrm{ LSB rms} $$

Since the resulting noise is still well above the LSB, cutting the average back to the original 24 bits appears okay -- while still preventing potential overflow. This example uses a power of two for the downsampling factor since the division for the average is a simple right shift.

Downsampling by more than a factor of about eight (with this simple filter) risks getting too close to the Nyquist frequency for the 1-kHz passband.

Averaging fewer samples should alias less noise into the passband, but if you then have to truncate low bits to meet your bandwidth limit, you might end up with an LSB that is greater than the noise floor, which is bad.

If you have enough processing power at your transmitter, the best way to do this is with a lowpass FIR decimation filter that preserves your band of interest while avoiding the aliasing of noise.