Hi Matt. Thanks - mirrorless is a little different to DSLR isn't it?
Different yes, but not necessarily better (any more) - it heavily depends on the mirrorless. My best low light camera is the Nikon Z 6, but not by much - because the D750 is still so good. The most surprising camera in that respect, however, was the rather humble Nikon Z 50 - it punches way(!) above its class in terms of low light performance; it's almost there up until ISO 3200 when compared to FF, even the Z 6. But - almost, not quite. However, the Z 50 outperforms my former APS-C champion, the Nikon D5500 I used to have. So, it's not the "SLR" part, it's the sensor - as a whole, not just "bigger is better". My
Sony A7 II performs no better up to ISO 1600 than my A6000 - and both are outpaced by the Nikon Z 50. That's one of the reaons I'll sell the A7 II ... once it's feasible again around here.
That said, the latest

20MP sensors are still the best in that system when it comes to low light shooting. ISO 1600 is really usable, no real qualms. It's not APS-C class - but close enough - so close that I'd actually pick my E-M5 III over the
Fujifilm X-E3; performance is certainly good enough, and I.B.I.S. does the rest. And remember, your E-M1 II sports the same sensor!
All in all, you should be fine with the E-M1 II - it's a great camera, especially for the price. To my eye, it has no major weaknesses as a main camera - except in extreme cases.
M.