It sounds like you have a good handle on exactly what is going on then. Regarding the mains supply ground, I can well understand that you're stuck with it. Hopefully you don't have issues with non-bonded plumbing etc.
Personally, I would treat the non-earthed output mains adapter as grounds for return to Amazon, if the output is floating and leaking like that then it suggests that the IEC earth pin isn't actually internally connected. A dodgy sign in my book, who knows what other liberties have been taken inside.
![Undecided :-\](https://www.eevblog.com/forum/Smileys/default/undecided.gif.pagespeed.ce.cCZ6w-jWpR.gif)
You would almost certainly be better off with a second-hand major brand name brick off ebay.
For the bench supply, in your general situation, you might want to consider providing a ground lead from a chassis screw so that you can earth the output when you wish/need to do so.
Regarding the TS100 itself. The connector barrel outer (negative) contact is AC decoupled to the tip via a 100nF capacitor (C26, schematic attached). I would have thought
that would have shunted quite a lot of the (typically 2200pf) Y cap leakage current. I don't know what the voltage rating of C26 is, it's an smd part. There is bound to be some parasitic capacitance between the tip metalwork and the thermocouple / element and a little leakage current too. That
might be enough to unsettle the thermocouple sensing, but with C26 shunting the majority of the leakage current, I'm surprised.
Just a though, try checking for DC continuity between the connector barrel and the tip - there's always the possibility of a faulty tip I suppose.
In my iron (with and earthed brick) I have bridged across C26 to provide tip earthing without the inconvenience of a separate earth wire. There is a small voltage on the tip as a result of cable resistance, but it is very low. Obviously not something that you can consider until your PSU leakage issues and stability are under control.
One other suggestion that
might work is to change the firmware to Railm's. It uses different (uses more CPU H/W features than the stock f/w) the difference in the control loop might be more resilient, I don't have great hopes of that fixing the problem though. [EDIT:
https://github.com/Ralim/ts100 ]
You can probably tell that I am running out of ideas at this point...