How does transition width of a low-pass filter affect a message which is well within the pass-band?
For an FIR filter there are 3 main components that determine the filter length for equiripple designs:


Passband ripple
Stopband attenuation level
Transition width (Width from the edge of the passband to the edge of the stopband)


For other filter designs the filter order may be related to flatness of the passband and the rate of fall-off in the stopband (and initial stopband attenuation). The transition width is often expressed as a fraction of the original sampling frequency.

In your case the original fraction is $\frac{100}{1.92\times10^6}\approx 52.1\times10^{-6}$ which is very small and is going to require a very long filter. Depending on the algorithm used to design the filter, the algorithm may run into numerical difficulties. As an example, designing a 2000 tap FIR equiripple filter using Matlab's Remez type routines is often problematic.

Increasing the transition width by a factor of 10 will roughly reduce the filter length by a factor 10 as well. This is just a rough approximation and results will vary based on the design routine and the filter specs being used.

If you really need such a small transition width, it is better to perform you decimation in steps, e.g, break it down to decimate by 10 and then by 4. In the first decimate by 10 you can use a wider transition width. The transition width in the last filter will be the same as the original requirement, but because of the reduced sampling rate the filter length will be much smaller. Crochiere and Rabiner's "Multirate Signal Processing" goes into more details.