Mine was the same year 2008 as yours. I bought it in march of 2008 and sold it after my 2nd failed back surgery late 2011. I ended up in a wheelchair for almost 5 months and thought for sure my riding days were over. Only the passed 3 years I'm back to riding. Man do I miss that bike. Hands down the best street bike I ever owned. And I've had many over the years. But my heart is riding dirt.

 

It sounds to me like when the tank is full there's enough pressure (head) from the tank to push the gas around whatever obstruction there is. So, imagine a kink in the gas line, at first with a full tank it can force gas past the kink fast enough to allow the bike to run. As the level of the tank drops there's still some gas running past the kink but not enough to keep the bike going. The bike is using fuel faster than the tank is supplying it.

When the bike stalls and you go to restart it the level in the bowl is low. The cranking engine doesn't pull enough air to gather gas up and the bike wont reach running mixture.

There's still a little fuel making it past the kink, but not enough to run the bike. So, when you let it sit the bowl fills back up. In this case it'll run until the bike draws down the fuel in the bowl faster than the weak flow can replenish it. It would run a shorter time at full throttle than at part throttle.

I would disassemble until you find the blockage and also check the float level on the carb.

Always asked "what changed" when trouble shooting. Also check the lines for flow by forcing compressed air through them if you can.

The stock setup on those fuel lines are weird, if I recall right, with both lines going around some wiring or something in a way that makes it hard to lift the tank out. It's a PITA.

The TM takes fuel from a different position than the stock CVK IIRC. This may be the source of the problem if the stock line was rerouted. It seems to me the kink or blockage will be below the petcock as it effects both reserve and main.

Check the visible lines first to maybe avoid a bigger-than-it-needs-to-be job.