On further thought, I wonder if I could do this a bit differently.
My original plan was to have two independent four-quadrant channels available that can both source/sink current at either positive or negative voltage, with respect to a shared common. That would then allow either NPNs/N-channel FETs by using positive voltages on both channels, or PNPs/P-channel FETs by using negative on both. Two-terminal devices (primarily in my case LEDs) are easily done on just one channel.
But if all I need to do is transistors, a simpler arrangement might be had. Instead of two independent channels, this could be arranged as a single-channel device with a voltage sensor across it and current sensor at both ends, with an additional connection that can source/sink at some voltage inbetween the two. This additional connection can then supply base current or gate voltage.
It now simplifies my problem somewhat, as now everything will always be positive with respect to the "lower" of the main channel terminals. The main channel is a double-ended current sensor with a shunt resistor at both ends (for which sensor chips like the INA226 will do well), having a differential voltage meter across it. In addition, the third terminal needs bidirectional high-side current monitoring, and needs its voltage measuring. Ideally I'd still like to be able to do this one differentially, by being able to either measure voltage above the lower terminal (for N-type devices), or voltage below the upper one (for P-type).
This feels like a more workable setup that reduces requirements on trying to find dual-rail meter chips.