Low Pass Filter that can quickly turn on or off?
The easiest and most effective way of doing this is not to filter your signal but start taking plenty of ADC samples as soon as you hit the area you want to sample. Carry on sampling until the pulse is about to change and stop sampling.

Average your ADC samples - that's all you have to do and if you can't take "enough" ADC samples, get a better ADC. Alternatively, if you can use samples on the next peak, accumulate values and average over several peaks.

There are analogue solutions but these involve comparators and tunable low-pass filters because the filter has to be in-circuit all the time and all you are able to "move" is the cut-off frequency (when the signal reaches the bounds of what you want to sample). Use the power of a decent ADC.

Moving the filter frequency - use a switched capacitor filter and bring down the clock to create the required low pass filter when the signal has stabilized.

Alternative idea - Use your signal (once it has achieved stability) to charge a capacitor with constant current (V to I converter). The voltage attained on the cap divided by the time to get to that voltage is dv/dt - you know the cap value and therefore you can calculate what the average current was over that time period (I = c dv/dt). The average current is the average voltage applied to the V to I converter. Timing and sequencing is necessary to start the V to I then when you have taken your reading you need to discharge the cap to zero before the next cycle begins. It's called an integrating ADC - look it up.