How to compare many bandpass filters (for their signal propogation delay)? Let's say we have an audio signal sampled at 96 kHz, and we want to compare several bandpass filters to find the one with the lowest rectangular-envelope-propagation-delay (see graph below).

The band is interest is $[f_0-30\,\text{Hz}, f_0+30\,\text{Hz}]$, with $f_0 = 1000 \,\text{Hz}$, and we aim for an attenuation of, say $-50\,\text{dB}$ at $1300\,\text{Hz}$.

Is there a rule of thumb or a ready-to-use calculator (either as online website, or in Python) to find the best filter with the lowest propagation delay, for example in the case of a signal modulated by a rectangular envelope?

Of course I can run tests in Python, with many filters, and many variations of these parameters:


FIR filters (how many taps?)
Butterworth filters (which order, ...)
Elliptic, etc.


but it's a bit long, and a bit random.

Example: here with a Butterworth of order 2, we have a delay of about ~ 25ms for the propagation of the envelope (the x-axis is in ms indeed).



PS: is this more or less close to group delay?
For a linear phase FIR filter, the group delay will be about the same delay as what you have defined to be a rectengular envelope delay. 

For a nonlinear phase IIR or FIR filter (like the Butterworth), the envelope delay is associated with the dominant time constant of its impulse respone $h[n]$. 

Indeed what's happening in your plot is that the filter reaches its steady-state response, after the transients decay. And that would happen after a few time constants. For the relation between group delay of the filter and the time constant, it can be said roughly that, short transition bandwidths in the frequence response (sharp filter or narrow band filter with large time constants) yield more aggressive phase changes which also yields larger group delays at the passband frequencies.

A minimum-phase filter would yield the minimum delay (among the same frequency response magnitudes). A minimum-phase filter's impulse response has the fastest decay and begins more aggresively than the other filters; i.e., it's energy is concentrated at the very beginning of its impulse response samples.