A question on transforming a non-inverting comparator to an inverting one?
I suspect that this has something to do with the feedback loop (connected through R5 and R10).

Your non-inverting comparator looks like:







simulate this circuit &ndash; Schematic created using CircuitLab

and your inverting comparator looks like







simulate this circuit

In the non-inverting comparator, you have positive feedback: when input > bias, the output swings high, which makes the input even higher! That way, the output voltage stays where it is.

However, in the inverting comparator, you have a negative feedback loop: when the output goes high, it drags the input voltage a little bit lower, which makes the comparator turn back off. This is the "ringing" that you see.

All you need to do (as I mentioned in the comments) is take out this feedback - a comparator doesn't need it. This should remove your overshoot.