Controllers should know this better than us, we drivers tend to try and do all the stupidities
Not generally an ATC thing - we try and accommodate a pilot's requests but whether the pilot has the right/authority/whatever to make the request is down to him or her.
But many moons ago when I was new to this business I did a stint ad Edinburgh where the transmissometers for the main runway sit between the runway and a little stream - or burn as they call them up there.
It was quite common, especially on lovely gin-clear nights, for shallow fog to form over the stream and then drift slightly or flow until it covered one or more of the transmissometers. Result - everyone can see for miles but RVR in the low hundreds.
At that time the rules had recently been changed and prevented visual approaches on another runway - can't recall if it was ATC rules or flight ops rules that changed - but it was a sore point for many of the Trident pilots that ended up holding or diverting when everywhere around was crystal clear.
OTOH, I have also seen the real effects of shallow fog and slant visibility from my perch high up in the tower and the resulting, rather scary go-around.
Don't know what the current situation is with JAR/EU-OPS, I'm afraid.