There's no problem, in principle, in using external phantom power units with the Zoom. And there are dual channel models in additional to single channel ones, which might be more convenient in your application. There are also both battery and mains-powered versions... (and some will do both).
I'm afraid I have no experience of the Neewer models. Phantom power isn't exactly difficult to do, and it is a common-mode design which means that even if the power supply is noisy, the noise shouldn't
get into the mic signal... but that's reliant on having very
closely matched feed resistors. If the source resistors aren't accurately matched (and we're talking 0.5% or better, here) then the power supply noise wont appears as a common-mode signal, and some hum or buzz will be added to the mic signals. The cheaper the unit, the less likely that the manufacturer will have bothered to check the resistor matching.
So, there's cheap and there's cheap... The least-costly model I've tried and would recommend is the ART Phantom II Pro (about £60 if you shop around) which can be battery or mains powered, and has two channels in the one box... which would seem more appropriate for your application. https://artproaudio.com/product/phantom-ii-pro-dual-ch-phantom-power-supply/
Personally, I use a very high quality Canford Audio phantom power unit when I need to (mostly for testing review gear), but that's around £200...