It depends on the quantity of the booze in your sauce. If it's a predominant flavour then stick with good whiskey. If it just adds a little accent in the flavour you'll get away with it.
My current favorite BBQ sauce recipe includes a nice glug of Jack, but last time I made it up I had run out of Jack and the only whickey I had to hand was Bushmills, used the same quantity and there was only the slightest of detectable difference despite the two whiskeys tasting wildly different.
To be honest cheap Tesco Bourbon isn't that bad anyway. I don't rate standard Jack Daniel's it's just got the branding, their Gentleman Jack is another matter mind you. However if you were ever caught putting Gentleman Jack in BBQ sauce I think there's a few on here who'd hunt you down and kill you