Bulk Capacitor Ripple Current as a Function of Capacitance
You are correct.

The capacitor and the source impedance (R2 in your diagram) form a current divider. The equivalent conductance of the load (seen as a Norton source) forms another branch of the current divider that you haven't shown in your diagram.

If you reduce the impedance of the capacitor without changing any of the other components in the divider, then the AC current through it will increase.

But usually you only care about capacitor ripple current if it is causing excessive \$I^2R\$ losses in the capacitor itself, or if it is causing excessive voltage ripple. If the first is the important concern then it's more important to find a capacitor with lower ESR, or use multiple capacitors in parallel to reduce the ESR. If the second, then you should consider both ESR and capacitor value in choosing a new part, but a higher value capacitor may in fact be part of the solution.