Ok, problem solved thanks to @tjb_altf4. Turns out I needed to add the wireguard network to LAN_NETWORK. This is found in VPN Manager and is under Local tunnel network pool. I added 10.x.x.x/24 to my existing LAN_NETWORK value, separated by a comma.
Goal: To be able to access my server remotely via VPN while having specific dockers run through Mullvad VPN for their traffic.
What I've done so far: My server is running remotely and I have followed JuanMTech's VPN guide to allow me to access my server from home. Basically, it uses a baked-in Wireguard plugin and DuckDNS as the DDNS for ease-of-use. I have attempted to set up several of the dockers that have VPN support built-in and have been able to get them to run through my Mullvad VPN (verified by curl and ifconfig) but I am not able to access their WebUI while they're connected to the VPN. I have verified that LAN_NETWORK is set to 192.168.68.0/24 (matching my servers LAN IP range).
Is my goal achievable? Am I missing something?