Honestly, "bless his/her heart" or "bless him/her" or "bless his/her/your soul" or any rendition of that seems to be American, as far as I can tell. At least, those phrases aren't generally in widespread use across Canada. I talk to people from the Southern US every day, and I hear those phrases very often from that area from all sorts of people, so I never really considered it a Christian phrase so much as a cultural one.
That being said, if I want to de-intensify a complaint, or criticism, I'd probably gravitate towards something like tacking something like this onto a sentence: "whoops", "oh well", "poor guy", "but nevermind, that's okay" -- it shows I sympathize with the person, or feel for their mistake, and don't really mind it. Opportunities for that come up often at work. Someone will call in needing help after making a (multiple) silly or embarassing mistake, I'll have to report to a higher-up on it, and my explanation will generally be something like "I was talking to Mr X, the poor guy accidentally did 'Z' again. No prob, I took care of it".
The "bless his heart" seems insulting to me, kinda. I know it's not really meant that way, but it almost seems to say "he needs extra supernatural protection because he is too silly to do something simple on his own". I figure there's no need to invoke the supernatural..... we humans will be just fine, even though we make ordinary daily simple mistakes.