If you are afraid you will be traced, you have to consider this information of yours as it's possible to trace from an exchange to any bitcoin wallet you will send your bitcoin. You can use a bitcoin mixer but it does not guarantee you'll be untraceable, AFAIK, it will only delay the tracing as it will go through the different addresses which makes it complicated but it's not full proof that it's untraceable.

I think a mixer is just the best way for you if you would still decide to send that bitcoin to your recipient.