Mic-ing is generally a bad solution because you are at the mercy of terrible venue acoustics and engineers being bad at their jobs. Most fingerstyle professionals just stick with a really good pickup + preamp + amplifier setup but that would easily go over $700. The one that you can buy last in that chain is the amplifier, so you're best going with a pickup + preamp and using the venues' PAs for convenience and reliability.
I've never had any experience with magnetic swappable pickups so I don't know how good/bad they are but I've seen a few full-timers perform well with them. Most pickups can sound convincing with some EQing and tweaking on your preamp anyways. Your most preferred, cheapest option now is probably an M80 (or similar) + Zoom AC2 or AC3 (their latest models). For the magnetic pickup though, definitely look into A Little Thunder & their acoustic mounting ring which adds an octaved bassline to your playing if you aren't prejudiced against gear designed primarily for electric guitars.
https://www.alittlethunder.com/product-page
https://www.zoom-na.com/products/gui...oustic-creator