As an ethernet client and as a Wireless Hotspot. The ethernet host needs to provide DHCP to BomeBox. As a Wireless Hotspot and Ethernet Client the connections will be separate and may get quite confused if you are trying to access via both ethernet and Wireless as they will be on different networks but with the same BomeBox Name.
As a Wifi Client and Ethernet Client, they should both work and on the same network however again there might be some confusion because ethernet host might provide a different address to the ethernet as to the WiFi so best to use it one way or another or set up fixed IP on both sides.
Unfortunately if your Mesh insists on assigning DHCP addresses, it may incorrectly assign a DHCP address to something with a fixed address, so to be safe you should only connect your BomeBox as a WiFi or ethernet client but not both. In either case, they would be on the same network either way so you should be able to have access to BomeBox whether ethernet client only or WiFi client only.