It is a common misunderstanding that signatures prevent tampering. They do not prevent it, they only reveal it.
If a user is complicit (they intend to use a cracked/improperly licensed version), allowing them to discover the tampering doesn't get you anywhere.
Your best bet is not to use the unique fingerprint to block operation of the software, which will surely just lead to a warez version which has the license checks patched out, but merely collect data on multiple installations which may be used when pursuing legal action in a civil suit for unauthorized copying.