Hello,
To be honest once a wash care label is cut, the end of it will start to fray after repeated washes, and tumble drying. Not much can be done about that unless you maybe do some stitching of some kind? We have never been asked this question so it maybe interesting if you let us know your results here - if you are going to be removing the logo side of the label.
Many higher end labels that source our garments do just leave in the label, or have a labelling service add in another label over the top, or indeed remove our label completely. There are many ways to skin a cat...not that i'm advocating that
Do Continental washcare labels fray etc. when you cut the Centrefold label in half to take off the CCC branding? 
