Wally writes:

Also any thoughts on if using bleach (other than the problem
with toxic fumes) would damage the tile surface over time?

Bleach? Why? You misundertand the chemistry. You're not going to oxidize
this type of dirt.

If bleach worked at all, it is because is so strongly alkaline to preserve
the chlorine content.

You'd be better off with an alkaline butyl cleaner, like 409, Fantastik, or
ZEP industrial, and scrubbing.

But I suspect you're grinding and embedding material into a relatively soft
ceramic bisque, in which case, nothing is going to do a great job of
cleaning it.

Flooring, like parents, should be chosen carefully.