Does this
断りの電話をかけさせないで下さい。
roughly mean "please don't make me hold your phonecalls"?

Pretty sure it doesn't, but it would help to have some more context.

I've looked it up piece by piece on alc.co.jp and I think it means "Please don't make [me make] calls of negative replies."

As in, "Please don't make me call back and say, 'no.'"

That's about what I would have guessed too, but it doesn't make a whole lot of sense without seeing what leads up to that sentence.

sorry - should have put it in context. It's a secretary complaining to her boss. I thought it was literally something like "please don't make me refuse phone calls".