I am assuming you will be staying withing range of wifi/bluetooth per your description of a 4x4 meter square.
If so try QField with rover RS+ streaming location via bluetooth onto device. You can set your project to any coordinate system you want and will project on fly from WGS84 Reach RS. You could try writing your own coordinate system file and bring into QGIS defining your own origin and parameters. But for your needs this should get you pretty darn close.