The rule I've run across (and I don't know that it has a name) is that you'll usually want about 750 words times (characters plus locations) in your story. You can get away with telling rather than showing a bit more in short stories, so if you can cut the backstory down to a sentence or two, it could work.

Is there a way to weave the backstory into the story? Have characters reference it?

Instead of talking about a guy making a mistake and killing his girlfriend in a scifi heist, you could reference it, and the audience would get the point. As a sleep-deprived example,

"Hey, John, any way you could kill the power, like you did in Denver?" Mark leaned around the corner and fired another laser at the robots.

John held his head down. Jessica danced through his mind, and he couldn't stop tears from flowing. Never again.

"No," John said, "too much of a chance they could go berserk before they shut down."