Normally this type of flooding is due to the floats in the bottom of the 2 x carbs not shutting off the petrol when the bowls have the correct level of petrol in them. If the bike has not been used for a while, the petrol in the float bowl can become a bit gluttonous and cause things to stick. I'm not sure how much clearance there is under the carbs but I would be tempted to remove both float bowls with the carbs left in place to see if you can cure the problem. Like most things on the F650, having to remove the carbs entirely to properly clean them is a bit of a hassle.
Please see the attached YouTube video to see what's involved in solving the flooding problem.
2006 F650GS in Black