You can build a rifle legal in both but it will not be ideal for either.
Spot on!
........ and just to add to that, both disciplines see such tiny margins in results in top-end competition, whether group size or score, that competitors now have to strive for perfection, a single additional lost point in F over a weekend's multi-match fixture often changing final results by several places in high-end matches. The GB F-Class annual European Championship meeting earlier this month saw 3 points (out of five matches, 80 shots and 400 possible points) cover the top 10 shooters' aggregates, and of these, places six to 10 were identical on score, but determined on V-count. (Xs to US competitors.)
"You can build a rifle legal in both but it will not be ideal for either." I think Mr. Wheeler has it right, but I bought a used F open rifle with a Panda action in 6 dasher that will hold its on at 600 yd F and since it weighs a smidge less than 17 lb is legal for BR. So from my perspective it's just more opportunities to shoot.
I too have done this and enjoyed myself, which it should all be about at the end of the day, but you have to accept that the results may prove limiting in some circumstances and weather conditions, even at club match level.
Maybe we (in the UK) have windier ranges than you, but the sixes rarely do well these days even at 600 yards in F-Class. (Of course, we don't string-shoot, rather shoot alternately in pairs or even threes with a mandatory 45-second allowance to re-read the wind and take the shot.) It's so unusual for a six to win at beyond 500 yards these days that it caused considerable favourable comment when one of my Diggle fellow shooters took the F-Open class with a 6-Dasher in a 600 yard match a couple of weekends ago.
Another factor is some of the actions used in F-Class which are not optimised for fast operation and trouble-free feed. Again, likely another UK orientated characteristic, but especially in the early days of the discipline, the four-lug SWING / Paramount / RPA series of actions were widely used. Excellent actions though they are, and capable of producing superb precision, their 60-deg bolt throw with hard lift and 'notchy' feel make them pretty hopeless in BR where shots need to be got off quickly with minimum disturbance to the rifle position and aim. Three lug actions such as the Barnard 'P' have become the F-Open norm here, but one doesn't see this type too often in BR either.