I have a vague plan to get to a LeJog one day soon (Paris first next spring)
November to April is really in the grip of winter - too dark and cold.
July and August would be a pain in the tourist areas and (although last couple of years not so) could be a real heatwave.
So you end up with May/June or September/October
Monthly averages
May
Avg Low: 10°
Avg High: 18°
Avg precip: 3.89 cm
June
Avg Low: 13°
Avg High: 21°
Avg precip: 4.64 cm
September
Avg Low: 13°
Avg High: 20°
Avg precip: 4.97 cm
October
Avg Low: 11°
Avg High: 16°
Avg precip: 7.07 cm
Above is for London (I know not even on route) but October looks to be much colder and much wetter than September.
May comes out as nice and dry but may be a bit cool.
The other thought is in training. Will you be up to it in May (most of us cycle less over the winter as there is less opportunity) whereas you should be at a peak by September having had summer to build up nicely.