How can I add wiring to get my three-way switch to work properly?
That kitchen switch loop needs to be redone to meet Code

Right now, the reason you are searching in vain for a neutral for your bathroom light is because you violated NEC 404.2(C) when you wired the switch loop for the kitchen lights by capping off the red and using the white as the always-hot wire.

You need to fix this as follows (with the breaker off, of course):


Remove the black and white wires going down to the switch box from where they were connected in the kitchen light box.
Nut the black wire from the switch box with the black incoming hot from the power source and any other always-hot wires in the kitchen light box
Nut the white wire from the switch box with all the neutrals in the kitchen light box (including the white pigtail from the kitchen light fixture
Connect the red wire from the switch box with the remaining terminal on the kitchen light fixture and any other switched-hot feeds that are present in the kitchen light box


Moving onto the switch box:


Unhook the black and white wires going up to the kitchen light box from whatever they were connected to in the switch box
Connect the black wire from the kitchen light box to the black wire going off to the other switch box, and to a black pigtail going off a brass screw on the bathroom switch
Connect the red wire from the kitchen light box to the differently-colored (likely black) screw on the kitchen light -- it should be the only open screw
Connect the white wire from the kitchen light box to the white wire from the bathroom light box
And connect the black wire from the bathroom light box to the remaining brass screw on the bathroom switch


Once you're done with this, you can button things up and enjoy your new bathroom light!  (If you want advice on the mudroom receptacles, by the way, edit your question with more info and I'll incorporate fixing them up into my answer.)