Designing a Fast Powerup Bandpass Filter
Your problem is related to &quot;integrator windup&quot;*. It's what happens when a linear system goes non-linear (in this case by clipping) and stuff like phase and impulse response are out the window. At this point the integrating components in the feedback loop (ie, caps) will integrate an error signal which is garbage and it takes a while to get rid of.
Optimizing the filter impulse response for quicker settling will help shorten the settling tail, ie what happens AFTER it comes out of clipping. But when it is clipped it is no longer linear, so optimizing its linear behavior is not useful.

I have tried adding a FET between IN- and IN+ controlled by the MCU at powerup.

The purpose of the FET is to charge the cap. Here's an example for the non-inverting circuit:

Input is a step on top of a sine. Blue is original, red uses the FET to charge the 220n cap at powerup. It settles in less than 100µs, assuming the source has low impedance (I put in a 100R resistor for source impedance). Since MEMS microphones have internal amplifier, I expect lowish impedance. Note you don't have to use a FET, a microcontroller pin will do (switch to output 0 then high-Z) although it might inject some noise in the signal.
However I don't like the non-inverting circuit because it only processes half the peaks and will clip on the other half. Also if the opamp's offset is of the unlucky polarity, you get no output signal as long as input amplitude is below offset.
Inverting configuration:

Here a LED (or any diode with a Vf a bit below Vcc/2) shorts the 147k resistor when there is a bit too much voltage on it, which charges the 470nF cap a lot faster. If you put 2 diodes in antiparallel, it will also work for the other polarity and quick-settle instead of clipping after any step at the input... at the cost of a bit of voltage headroom on both sides.
Note shorting both inputs of the opamp with a FET will only make it clip, either up or down depending on its input offset voltage. So that didn't work. You have to short the high resistance that makes charging the cap slow, ie the 147k resistor.

That seems nice. Note I used a Cheat-FET with a magic driver courtesy of the simulator. You'll need a real FET switch. You could even have the micro look at the ADC samples, and if they look like the opamp is clipping, have the software flip the FET switch to help it come out of clipping much faster.
The 4k resistor still limits the current which still keeps it slow. So let's move a wire and let the switch short it too. When the FET is ON the opamp should still work pretty much as a follower.

It works, and it needs a much shorter ON-time on the switch, like a couple hundred µs. That should solve your problem.
Doing the same on the multiple feedback topology would require one extra switch since the two problematic resistors don't share a pin so it's not possible to short both with one switch.

= Integrator Windup is what happens when a badly designed audio amp clips:


Green is what it would look like without clipping. Red is actual output (well, actual as hand-drawn). Just because the amp is clipping doesn't mean the capacitors at various places, and especially compensation and feedback, stop processing the error signal (hatched area) which gets integrated. So at point 1 although it should do its best to go out of clipping, it won't decide to do so until that integrated error signal (ie, charge in caps) has been purged, which occurs at point 2 when the amp goes from nonlinear back to linear. By then the point where the output should be has moved quite a bit, so it immediately goes back into nonlinear (slew rate limiting in the other direction) which sometimes involves saturating a BJT, so it overshoots, waits until said BJT goes back to linear, and finally resumes linear function... If the output stage is rail to rail, and with a bit of luck, the lower transistor will conduct while the upper is still fully saturated, shorting the supply, and while you stare at the scope screen, smoke happens.