Any way to patch cracks / chips in pottery butter dish?

My wife has a butter dish that she has had longer than she's had me.
It's some kind of pottery with a glaze finish. Made in Italy.

Recently, a small piece of the finish chipped off. It's in the middle
of the base where the butter sits. The finish is shiny white. The
material underneath, now exposed, is a dull grey.

The chip is about the size of a pea and triangular in shape. I do not
have the chip. (Probably had it on a piece of toast.)

If I look closely, I can see a series of hairline cracks in several
across the base.

Is there any way to patch that chip? It doesn't have to be perfect,
because it's covered most of the time, but it has to be pretty good. I
was also hoping to stop further chipping.

I asked over on the home repair ng. The suggestions there were
appliance touch-up paint, acrylic paint, and tinted epoxy.

I bought the touch-ip paint and the acrylic paint. The acrylic paint
was the wrong shade of white. The touch-up paint was close to the
right color so I tried that first. I filled in the chip divot (only
took 2-3 coats as it was very shallow) and touched up the hairline
cracks.

When it dried, I polished it with polishing compound used for car
paint. The polishing removed all of the paint from the cracks. It
looked like it didn't stick to the glaze at all. The paint in the chip
is there, but I was not able to get a nice smooth finish. It's not
bad, but not great.

Is there any way to repair this dish? It's not valuable, but she likes
it.

If there is no way to patch the chip and cracks, can I repaint the
bottom? The dish has a lip running around the base. I could paint just
that portion if I could match the color fairly well. I supposed I'd
have to roughen the glaze to get it to stick, so I don't want to even
start on that unless it's going to work.

Any ideas appreciated.