It sounds as if your HP laptop brick is simply not up to the job and is going into an over-current protection /shutdown cycle. It is probably right on the limit with the original bit and just over with the new ones, especially when the tips are cold and it is running at 100% duty cycle.
If you look at the TS100 manual, it specifies a PSU current rating of 2.1A for 19V supply (40W), so you're a fair bit short of that.
I believe there has been to odd occurrence of a faulty, shorted element but the fact that they both work fine with Lithium cells and the 4A PSU seems to eliminate that.
It would probably be best not to use it with a repeatedly browning out PSU, worst case it might cause damage to the switching FET or possibly even corrupt the firmware.
P.S. You're probably having no luck with your current measurement because the element is controlled by PWM switching and the duty cycle will be changing as the tip warms up.