Button batteries vs 9V headache
As a rule of thumb the V/ mAh impedance in an hour will depend on the energy density of the chemistry.  So size and weight matters for the effective series resistance and button cells are in the range of 1k which add in series while alkaline cells might be 1 Ohm +/-  an order of magnitude depending on size and quality.

So you have undersized your power requirements with excessive ESR, which the inverse defines your load regulation on open cell voltage.

You may be better off defining your mA max and mAh min requirements and look at LiPo flat packs.