Cheers Cyprusice, I owned the exact same bike up until a few months back, the gas tank gave me the biggest headache, constant build-up of rust flakes.
It could be the shape of the tank which is quite unique to that model but I had to remove it for cleaning heaps of times and like you the in-tank filter was long gone.
Also as someone else mentioned, I wouldn't let her idle for 15 minutes, that's way too long for an air cooled bike, you run the risk of fouling your plugs which will give an indication just like you describe.
If it was my bike I'd clean out the carbs, you will find these strange little orange plastic petrol filters actually inside the carbs, mine were clogged to death with rubbish.
Removing the Custom carbs is not such a hard job as it is on the older "Z"' model I have now.
The tank, I would possibly fit an inline filter if you can't replace the in-tank one.
Apart from that I found the 650 Custom to be quick and once cleaned out, reliable.
Best of luck.................Hush.
P.S. if your front brakes start binding and wont release there is a tiny tiny pin hole in the bottom of the master cylinder that you must keep open, mine tried to trow me over the bars when brakes locked.