Double NAT should be OK. I have done it a few times. Some things get upset but mostly not.
I would.
Make sure that you can go back to the old configs.
Assuming only a single external internet address is available.
Choose addresses and reserve them - say on a piece of paper.
DHCP is OK as long as they are static.
Lets assume you want to use .7 for this device and we assume a
mask of 255.255.255.0.
QNAP real address 192.168.0.7, subnet 255.255.254.0.
QNAP NAT on network 192.168.1.0 subnet 255.255.255.0, for your sanity I would choose the same host portion, say .7.
QNAP translated port on the outside. Can be same port as used inside or a different port.
On Dream machine set up a static NAT to translate
192.168.0.7 to 192.168.1.7 for all ports
On LiveBox set up Port Address Translation to translate
the internal NAT and port 192.168.1.x:QNAP-native-port
to whatever you want to use on the outside.
n.n.n.n:QNAP-outside-port
Dream Machine config - static NAT
QNAP real address 192.168.0.7
QNAP NAT on middle network 192.168.1.7
LiveBox config - Port Address Translation
QNAP mid network 192.168.1.7:QNAP-native-port
QNAP outside network n.n.n.n:QNAP-native-port
I would recommend changing the outside port to something unusual.
Sure, it's security by obscurity but I would do it anyway.
In that case:-
LiveBox config - Port Address Translation
QNAP mid network 192.168.1.7:QNAP-native-port
QNAP outside network n.n.n.n:QNAP-outside-port
NAT can be a complex and head bending thing and I have here left many simplifying assumptions unaddressed.
Consider putting all the addresses on the network diagram. I did a lot of NAT once upon a time and we did a separate diagram for each NATted service with all the addresses and ports on it and also with a table of all the addresses and ports in order inside to outside, with a technical description of each one.
PS Thanks for the diagram, I don't often try to answer questions where the questioner expects me to draw the diagram. I can't do networking without diagrams:-)