Another possibility. Not as good as the Tektronix system, but way more compact and good enough performance for what I need: a Lecroy CP015 (alternatives are CP030, AP015 and AP030)
I run this probe without a Probus interface, just with a +-12V power supply connected directly to the probe (and a BNC 90°-adaptor to connect it to the scope). There probes can be had for 300-500€ on Ebay, less if you are lucky. When power is turned on, the probe runs a degaussing cycle. So for I have not looked into trying to send I2C commands as these protocol is not well documented (and it works good enough without).
Test was run with the Yokogawa GS200 supplying a 30mA current (to mimic test conditions above). I did not log the current of the Yokogawa, but I am sure this is completely flat (max a couple of µA variation). Probe was connected to a BM869s and logged via testcontroller software. As the output of the probe is only 100mV/A, the signal was only 2.6mV. Temperature in the room was not tightly controlled, so some drift could be temperature related of the >4 hours this test has run.