I gave up on Flying Blue long ago after being very rudely patronised by a KL lounge dragon who thought she was far more important than the passengers she was supposed to be serving.
British Airways Executive Club seems to be a far more convivial set-up. The staff are infinitely more courteous and KLM's and the BA Galleries facilities are arguably some of the best available.
If you are indeed spending £10K a year you should attain BA Silver within a very short time indeed. Gold will surely follow!
[For the avoidance of any doubt, these opinions are my own and - moreover - I am not employed by British Airways or any of its subsidiaries.]