This has brought out a wicked thought; it's very tempting when inbound to ask a controller for the QFE by runway just to gauge their sense of humour. I suspect if it were a TWR, they'd have me orbiting for a few dozen minutes whilst they sent someone to measure it to the nearest Pascal for me. 
I've been out of active controlling for a good few years now but I fear you might be disappointed. As I recall, at airports with a precision approach, the QFE to the tenth of a hPa, and for each threshold if necessary, was displayed in front of the controller all the time. You may have been fooled because the controllers are smart enough to know which aircraft might want that level of precision and, the vast majority, that do not. Doubt that much has changed.
The only cause for a short delay might be while the controller wonders what the heck you're asking for.......or maybe says to his/her colleagues 'You're not going to believe this, but....'.
