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Would connecting the output terminals of the band-pass filters with each other cause some issue? Like affecting the low-pass corner frequency?

Of course it would.
All the output capacitors would be in parallel, giving the equivalent of one larger capacitance equal to the sum of the three.
You need to run each output into a summing amp with high value input resistors, to avoid interaction and any significant effect on the rolloff frequencies.

Note: U3B can't reliably drive a 33Ω resistor.

 

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