Hello @Sebastian Vatland Sonesen and welcome to Microsoft Q&A.
If you would like to make the failure of the copy activity, not fail the pipeline, then add a wait activity after the copy activity connected by an on-failure dependency.
If there is already an activity following the copy activity, do not add the wait activity. Simple change the dependency from on-success to on-completion.
When an activity is followed only by an on-failure dependency, this indicates you expected the failure to happen, and whatever follows is intended to handle the situation. In this way the pipeline will not fail.
Please let me know if this solves your problem, or if you want more explanation / help. Thank you.