You shouldn't redirect on your update route. You should return a JSON response at the end of your update method. If you use class based request validation (instead of a validation inside the controller function), validation errors will automatically get returned as a json response.
If you block form edition and resubmission as long the promise is not resolved, the easiest way to do so is by simply returning an empty JSON response which will return a 200. Since it returns a 200, it will hit your then callback (if using axios) and in there, you can put your logic to update your table row. Else, if an error is encountered, the catch callback will be triggered. You can then renable your form in the finally callback.
return response()->json();
You could also return your updated data in the json response if desired.