The fan controller is certainly a big part of the problem, Most fans in most quiet PSUs are actually quite noisy at full speed; it's the smart controller which keeps the fan from running so fast when it doesn't have to.
So yes, you could just stick a resistor in line to reduce the voltage to the fan (which I think for that series was still a 2-line simple voltage controlled motor). How quiet (and smooth) this sounds will depend on the fan of course, and you might have to play with the exact resistor value to get the right speed -- ie to have it turn on consistently.
If replacing the fan altogether, you want something with a known low start speed and not more than ~1500rpm max speed. Perhaps 1200. But this depends partly on what the default min voltage to the fan is. You might want to measure that (across the fan terminals with a multimeter) to be sure. If you choose to do so, take great care with exposed wiring, as a live PSU can certainly ZAP you pretty good.
Thanks for your helpful response MikeC. I suppose that if I was to use a different fan that was quieter but moved less air at minimum speeds, then if the PSU become too hot, it should increase the speed more. As long as the maximum speed was adequate to keep things cool, then I at least won't be any worse off.
I have read in another thread that someone used a 1200rpm fan, but I didn't want to just go by one person's experiences. So it helps a lot that you agree with that number. That person also said that the fan in the TX650 is 135mm rather than the advertised 140mm, but I've looked up the one he mentioned (which is no longer made unfortunately) and it did have holes the same as a 140mm fan would, so I can only assume that any 140mm fan would fit.
I think I would rather try a new fan first before experimenting using a resistor or external potentiometer to control things. To save me taking my PSU apart (which is quite an effort as I have all the cabling tied down very neatly inside the PC case) twice, I'll just find a fan that advertises itself having a low start voltage and hope it'll do, rather than taking it apart to measure that voltage first.
Off to do some fan research next!
Oh and thanks for the safety tip, it's always good to be extra careful with PSU's. I'm an ex electronics engineer so know what I'm doing, but I would recommend anyone who hasn't done such things before to take great care.