ARP message to enc28j60
In general, to have any kind of meaningful insight into what happens, you'll need a way to examine the packets on the networks. Both between the Ethernet gadgets and the Wifi AP, and between the Wifi AP and the PC(s) connected to the Wifi network.
On the PC(s), you can run a network traffic analyzer to watch the traffic seen by that PC, but your Ethernet gadgets probably don't have enough resources to provide a meaningful traffic dump. In that case, you'll need some kind of Ethernet sniffer between the Ethernet gadgets and the Wifi AP. The easiest solution is usually an Ethernet device capable of tapping the traffic (a plain old Ethernet hub, or a manageable Ethernet switch configured appropriately), and a PC connected it to receive the tapped traffic, and analyzing it with a network traffic analyzer (a software like Wireshark, for example).

Then you should examine the differences between the ARP requests coming from a PC directly over the Ethernet LAN, and those coming from a PC over Wifi, bridged to the Ethernet LAN. These requests will differ (for example: the Wifi AP will put its own MAC address into the Ethernet header).
Also, you could examine the 'good' replies from your Arduino and the 'bad' replies from your box to see the differences.