Overshoot when using Bessel filter
If it were a low-pass filter, I would say you need to turn the resonance (or $Q$) down enough that both poles are real.

What's the order of your bessel filter?

But a bandpass filter is necessarily resonant.  i might think that a BPF must have poles that are complex.  (now that i think about it, a BPF does not necessarily have poles that are complex conjugate.)

if the BPF has gain exceeding 1 (or 0 dB) for its resonant frequency, then then it should be expected that the output will be larger than the input at that resonant frequency.  but if the gain is less than 1 for all frequencies, i dunno why it would overshoot at that frequency.

but there are zeros for BPF also.  one at DC and another at Nyquist.  so maybe you have to turn down the $Q$ to $\frac12$ so that no pole is "ringing".