If the public DNS for home.domainzzz.com points to your SME's WAN then you should be able to install the letsencrypt contrib and let it generate a cert for that domain.
I don't think they can if 'domainzzz.com' points elsewhere.
The server will want a cert that is good for both 'home.domainzzz.com' AND 'domainzzz.com' - I have had this situation myself.
2 SMEs. One is the main mycompany.com and has certs for mycompany.com,
www.mycompany.com, mail.mycompany.com
I wanted a completely separate SME host called files.mycompany.com but If you ONLY get a cert for files.mycompany.com (which you can) the config will still want a cert for 'mycompany.com' as well.
You would need to hack the htpd config to remove references to 'mycompany.com' to get around this I think. It is a limitation of the way that letsencrypt/SME works at the minute.
(Note I could be completely wrong here !!
There is hack that you can employ. Mod the server to be someotherdomain.com and then set up a new
domain called files.mycompany.com and then ONLY get certificate for JUST the 'domain'. That will fool SME)
These should reveal a bit more information.
db domains show
db hosts show