The only variant left to try is:
"How close can you get to the Diamond lattice with two-sided ideals of a ring?"
Naturally, the commutative example in the first post is an example with six ideals, the Diamond with one ideal on top.
I'm putting (what I think is) the solution below for review. If all is well then it contains an alternate proof of why the Diamond can never appear in a lattice of ideals with $R$ at the top, even for noncommutative rings. (The previous proof factored $R$ into local rings.)
This brings the line of questioning to closure to me, but maybe someone else has a good variant too!