I'm guessing this is for some sort of NMR/MRI application.
First of all, accuracy of 1ppm... forget that, not even the super-high end amplifiers achieve that. They may have precision/repeatability around 1ppm (and even that's highly debatable), but not absolute accuracy. Static offsets don't matter (so long as they don't drift over short time scales), those get calibrated out at some point.
Even so, I'm not aware of any current sense resistor technology which can give 1ppm level precision, especially when including shunt amplifier drift (unless you have all of it oven-controlled). If I really needed to meet that spec I would go for a good fluxgate transducer.
For the power amplifier topology, it's hard to recommend something without knowing more about your load (resistance and inductance) and desired slew rate and duty cycle. Class G is an option I've worked on before, but it can be very difficult to get the rail transitions to behave nicely. You fine tune it, and it seems to work well, but then someone says "oh I need to adjust amplitude/slew rate/pulse duration a bit" and suddenly it acts up again.
I don't know what you're referring to when you say "switching the op amp feedback loop". In a class G circuit you only switch the supply rails for the amplifier, not its signal path.
I'm wondering why you're using a Howland current source, as opposed to a simple noninverting amplifier configuration (with the current shunt connected the GND, and the load being in the feedback path). AFAIK the Howland circuit is chosen in cases where the load absolutely has to be connected directly to circuit GND. But it has more error sources than a simple noninverting amp circuit.