I have already calibrated a model using SWAT+ CUP. To test it in SWAT+ Toolbox, what I did was copy the calibration.cal file from my SWAT+ CUP project and paste it into the calibration folder of the SWAT+ Toolbox (something like: ...\Scenarios\Toolbox\Calibration\calibration.cal), replacing the original file.
Make sure to remove all parameters listed in SWAT+ Toolbox so they do not overwrite your manually edited calibration.cal file.
Then, run the model in the Toolbox using the Run Model tab. If you run it through the Calibration tab, the Toolbox will modify your manually edited calibration.cal file.
This was the way I managed to run my model in the Toolbox using a calibration that came from outside the Toolbox. If anyone knows another way or if this is incorrect, please feel free to comment here.
Mario