You will probably need something beyond a CNC shield for a 3.1A stepper, The CNC shield takes plug-in driver modules of various kinds (e.g. A4988, DRV8825). They do the same job but their current capability varies.I think 2A is about the max though – the form factor probably precludes anything more.
To source 3.1A you probably need a separate driver which will likely have screw terminals rather than little square pins.
Using a lower current driver won't fry anything – the driver will limit the current but it may not provide enough stepper torque to do the job,
Sorry I'm not au fait with the higher current drivers because my smaller machine is OK with the little ones but hopefully someone else will come along to advise,
Control-wise you indeed could go for a microcontroller running GRBL but I don't think GRBL can support much of a control panel because input/output pins on the Arduino Uno are mostly needed to talk to the steppers. Sorry I'm not sure what hobbyist controllers and firmware can do this but maybe someone else can suggest something.
EDIT: typo
Edited By Andy Carlson on 18/04/2021 18:16:15