Electrical – Grounding a receptacle when there is no ground wire

I think you would be better off using a remodeling box (with the flanges that rotate out to grip the wall) and ignore the stud altogether. With a remodeling box, you just have to cut a hole big enough for the box, whereas if you use a regular box, you have to cut away more so you have room to nail it into the stud.

The main difficulties you'll face are that you don't know exactly where the lath is placed within the wall, and that the plaster will be brittle with age.

Start by using something sharp to gouge out a small hole in the center of the spot for your new receptacle; I use the pointed tip of a drywall saw. Once you've exposed the lath, expand the hole up and down until you find the edges of that piece of lath. The other lath are going to be approximately the same size and spaced up to maybe 1/4" apart. Adjust your planned spot up or down a little so that at least one edge of the box lines up with a gap in the lath -- it's easier to cut through the plaster on its own.

Expand the hole until it's big enough for the box, then use a jigsaw to cut away the lath. Vibration from the saw can cause more damage to the wall so take it very slow and easy. (Another reason to avoid the stud: if the lath ends at that stud, you'll only have about 1/2" of it to hold the plaster in place.)

"The bridge between the screws was not broken" Are you 100% sure of that? Because the common use for black and red wires and a two-pole breaker in a kitchen outlet circuit is for a common neutral circuit (multiwire branch circuit), and shorting the two phases together with the new outlet would produce exactly the symptoms you have seen. So I'm going to suggest that the two sides of the old outlet are not actually connected, and if you don't have a meter you can check that with, I'll provide you an alternate test that does not involve looking at the connection, which you've already done and come to (I suspect) the wrong conclusion about:

Cap the red (or black) wire and connect only the black (or red) wire to the old outlet. Plug something into one receptacle, and see if has power. Then plug it into the other side. Repeat with the other wire (only) connected to the other side (only.) If I'm right, in each case, only half the outlet will have power, and which half will change with which screw you connect to.

It appears that the device you have probably only has one hot screw, so it cannot be connected the same way as the old outlet with two separate hots; so leave one wire capped off, or install it somewhere other than your kitchen (the reason for using this type of circuit in the kitchen is to provide more circuits to the countertop outlets - this device defeats that if it cannot have two separate hot feeds.)