Integrate product of Dirac delta and discontinuous function?

The solution you have suggested is a perfectly good one. The only problem is that you are venturing outside the bounds of conventional Distribution theory. Consequently, the onus is on you to prove whatever properties of your definition that you use.

Actually, the Heaviside step function is more usually treated as a distribution itself. Bracewell's book, "The Fourier Transform and its Applications" considers the problem of defining a value at the discontinuity and concludes that it mostly doesn't matter.

What you are doing seems somewhat similar to the idea of density in physics. I'd be surprised if the physicists or the electrical engineers hadn't already confronted this issue. However, after looking around a little bit I am unable to find anything specific.