I pulled out a plot I made a few years ago showing breach width at mid height vs height of water in breach. (Data mostly borrowed from Froelich, D.C. "Embankment Dam Breach Parameters Revisited" in a 1995 conference. See link below for full citation.) The ratios ran from about 2 for small reservoir (<1000 ac-ft) to 10 for large ones (>10,000). I don't know how much the smaller ratios were constrained by the abutments (so they couldn't get any wider like could happen with a very long dam. Some of the higher ratios came from flow slides (e.g., Buffalo Creek WV). For small reservoirs, I drew a line at 2.5 H + 180 feet, which almost forms an upper bound. The best fit to the data for small reservoirs would be more like 2 H + 100 feet. (This is out the window, of course, if the abutments constrain the size of the breach.
The best source on breach parameters is:
That's a pretty thorough review of the subject.
Please do not use the prediction of breach occurrence in the Dewey & Oaks and Dewey & Gillette papers cited therein. It's been shown to be questionable, or worse.
As to the depth of overtopping to initiate breaching, you might pick 1 to 2 feet or even more, depending on the erosion resistance of the material and the duration. It is not uncommon for dams to be overtopped by 2 feet and survive, if they are constructed of clayey material, have good turf cover, etc. It's going to be a SWAG anyway.
I'm curious about how much difference there is in the peak Q between 1 foot and 4 feet of overtopping. Keep us posted please.