Designing digital low pass filter with low pass-band group delay
Minimum phase filters will not give you a near constant group delay. You can design a non-linear phase FIR filter with a linear desired passband phase with a specified group delay that is smaller than the group delay of the corresponding linear phase filter. If you use a least-squares criterion, this is equivalent to solving a system of linear equations.
As an example, take an FIR lowpass with $61$ taps. The group delay of a linear-phase filter would be $30$ samples. Now we specify a group delay of $20$ samples instead:

w = pi * [linspace(0,.2,100), linspace(.25,1,375)];   % frequency vector
D = [exp(-1i*w(1:100)*20), zeros(1,375)];   % desired complex frequency response
W = ones(1,475);   % unity weighting
h = lslevin(61,w,D,W);   % least squares design

The function lslevin.m can be found here.
The result looks like this:

Note that the passband group delay is of course not exactly constant, but the desired group delay of $20$ samples is approximated quite accurately over a large part of the passband.
Trying to achieve a low group delay in the passband comes with a penalty in the magnitude response. The figure below shows the magnitude and group delay responses of $4$ length $61$ FIR lowpass filters with different desired group delays in the passband. One is a linear phase filter with a constant group delay of $30$ samples, and the other $3$ filters are non-linear phase low delay filters with specified passband group delays of $25$, $20$, and $15$ samples respectively. Clearly, the magnitude response deteriorates with decreasing desired passband group delay.