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One way can be to disconnect one device for one or two days around that date and see if emails are still send. If yes, then proceed with the next device. Time consuming... But if you deal with that error every step brings you closer is a good step.
Another maybe better way could be to buy you a cheap managed switch (e.g. https://www.amazon.de/Netgear-GS105E-200PES-Managed-konfigurierbar-deutscher/dp/B00MYYTP3S?language=en_GB&th=1) and connect your LAN devices to that switch. Then you can connect this switch to your Router. After that you can define a mirror port for this Uplink and then you capture your traffic.
A third way is the proposal of @Graham or you can replace your router by a fritz box which also has build in capture functions.
