Hey Mike,
Yes, but this is a problem with the correct voltage too. This is due to the type of motor. You can avoid this by using an induction motor instead (also called spindle in this context), however this requires the use of a VFD. There are a lot of 100 V class VFDs on the market and a spindle runs very well with 100 – 120 V. It is another type of motor that will not get hot when driving low.
However I don’t think that a 10-20% lower voltage will make that much difference with the Makita router. The same thing happens when you lower the speed at the speed dial: The voltage will be reduced. A hand trim router with its universal motor is always a tool that you can’t leave running unattended and that can always burn your place down if you put too much chip load on it anyway, even if you drive it with the exact voltage. However I believe many people who show their projects here never drive the router to its limits, otherwise noone would use it in a CNC. And as explained above, neither the 50 Hz frequency nor the 100 V voltage would be something strictly prohibitive with such type of motor. A universal motor like this is in fact a DC motor that runs on AC. The commutator is the technical solution that makes the AC frequency absoutely irrelevant.
Sure! It is a hand router. According to the manufacturer, even the corded version is not allowed to be used in a stationary machine like a CNC.
If you manage to wire it to the battery contacts, that could very well be a solution. The good thing about the LiIon battery models is that they have a DC motor which is much superior to the one in the AC versions. I think in your situation I would try that if you don’t want to use a spindle (which would be the best solution).