To be clear, typical bench and handheld DMMs don't technically measure current directly, they measure the voltage drop across a known precise resistance (shunt) and calculate the current using Ohm's law. The reason the burden voltage is a "problem" is because these DMMs have a limit to how low of a voltage they can detect. Since V=IR, the resistance must be made larger if the current is smaller in order to maintain a voltage range that can be detected/measured. The DMM manufacturers make a design choice for each current range depending on the capability of the DMM. That is why lower-end DMMs typically have a higher burden voltage. (The 121GW uses an amplifier in some ranges similar to what the uCurrent does in order to reduce the burden voltage as much as possible.)
The absolute minimum voltage the BM869s can detect is 0.001mV (500,000 count mode). The lowest current it can measure is 0.001mA and 0.01uA. Based on the shunt values I provided earlier, we can see that 0.001mA results in a value of 0.001758mV and 0.01uA results in a value of 0.00101543mV. So both of these are clearly chosen to line up with the best the BM869s can do, 0.001mV. (10A shunt calculates to 0.00276mV.)
Another way to measure current is with a Hall Effect sensor found in DC clamp meters, but most are going to bottom out in the low mA area. Forget uA or nA.
Speaking of mA/uA/nA, I'll maintain there is no huge reason to need the uCurrent or similar since you can simply use your own known resistor as the shunt. For example, the uCurrent uses a 10k resistor for the nA range, so assuming you need to measure 1nA, the voltage drop across your shunt would be 0.01mV which the BM869s can manage well enough. It becomes trivial at say 50nA, which is 0.5mV.