The reason to use an isolation amplifier

Most of the power supplies that I have looked at have used some combination of analog and digital isolation, depending on the supply. Most of this is drawn from repairing a couple of HP/Agilent/Keysight E3600 series power supplies. Generally, there are at least two processors: one is an I/O processor for the GPIB/serial interface, and a different one manages the user interface and direct control over at least one power supply channel. These two generally communicate via a pair of optoisolators. The I/O controller shares a ground with the I/O connectors on the back. The main controller is grounded to the common of one of the power supplies. In supplies with more than one output, the other output is connected with analog optoisolators to the main supply. I think there was something like 6 optoisolators connecting the two channels together in my E3649A dual outut supply, some for analog signals and some for digital signals. I have a triple output Tektronix supply as well, but I can't remember how it's wired offhand. With optoisolators, I think they do a good job of passing an analog signal through if they are driven in current mode, so you will need op amps on both sides to condition the signal.

MCUs are very cheap these days; it may be advisable to put an entire microcontroller on each channel and then use some sort of serial protocol for communication through digital isolators. The Atmel XMEGA series has built-in DACs and ADCs and lots of serial ports (4+), for example. One of these per channel and one more for a main controller could be a very nice design, and it would only require two digital isolators per channel.

I'm starting to think that the only viable option is to have each channel galvanically isolated from the other channels.

Close, but I'd recommend that the channels can be tied together - it's the grounds which need to be galvanically isolated. You see,

(if neutral gets unplugged while relay is closed, the local metering IC ground will simply float up to 230 Vrms)

is terrifying. Have you paid attention to what you're saying? "Simply float up to 230 Vrms"? In other words, your entire system can be tied to 230 volts, and without your knowing it. Any competent electrical inspector will shut you down in a heartbeat.

You should condition your signals using either battery power or (even better) a low-voltage transformer whose primary is driven by AC on the measurement side. Use a separate amplifier for voltage and current, and if you really feel like you need to save on communication links, you can switch the amplifier inputs between your two sources. This will be rather peculiar, since your two sources are AC and DC, so I'd recommend a separate conditioning circuit for your 4 variables.

Once conditioned, you have a choice - either do a power-side A/D with optical links (photocouplers) to your measurement side, or provide analog optical transmission, such as by using a V/F converter, of your 4 signals, and acquire the signals using one or more F/V converters (you can easily put a multiplexer here). The first approach uses a minimum of A/D hardware, but it requires bidirectional optical links to control the ADC. The second uses more power-side circuitry (the F/V s) but the acquisition section is unidirectional. Your choice.