It's not internet connection changing that's changing the fingerprint. It's that the customer is disabling network adapters (when it adds no benefits and actually creates problems).
That being said, in TA 4.0 we've improved the fingerprint algorithm to use all real network adapters and enable disabled network adapters temporarily so that the hardware attributes of the adapters can be read.
So, short-term answer: tell them not to disable their network adapters.Long-term answer: use TA 4.0 or newer.