I can now answer the question, can you have two VoIP phones connected serially to the same Internet connection. Yes.
I still have my Vonage phone. The Vonage terminal adapter is acting as the DHCP controller. It’s connected to the DSL modem. From the Vonage TA I connect my Linksys router in bridge mode. That goes to the Packet8 phone, which, unlike the Vonage TA, seems to prefer being inside the router. The Apple Airport Extreme is connected to the second port on the Packet8. The Airport is also doing DHCP (I know, I know but it works) and uses an Airport Express in another part of the house as a WDS relay.
Both phones work. The wired and wireless networks work. The computer down the hall works. I’m surprised the whole thing doesn’t burst into flames.