I cannot give some advices to the specific problem asked in the question because I don't get further information from the comments. Maybe the wired connection with an additional USB-ethernet dongle is also connected to the same internet router than the wifi interface and there is a routing problem. But I will show how to use an additional wired connection to a different subnet.
First setup proxy arp as described at Workaround for a wifi bridge on a Raspberry Pi with proxy arp and make sure it's working. That's the same setup used in the question. Then just plug in the USB-ethernet dongle with an ethernet cable to the subnet. With this setup you have to ensure that the new subnet is another one than the wifi has. For example if wlan0 has got an ip address 192.168.50.182/24
by DHCP then eth1 could be set to 192.168.10.60/24
. Configure eth1 with this file:
rpi ~$ sudo -Es
rpi ~# cat > /etc/systemd/networki/04-eth1.network <<EOF
[Match]
Name=eth1
[Network]
Address=192.168.10.60/24
EOF
Then restart the network with the new setting:
rpi ~# systemctl daemon-reload
rpi ~# systemctl restart systemd-networkd.service
rpi ~# exit
rpi ~$
Now you can use interface eth0 as faked "bridge" to the wifi subnet 192.168.50.0/24
and you can use interface eth1 to connect the RasPi as normal client to subnet 192.168.10.0/24
.