Time Capsule (3rd Gen) seems to deny internet access to clients with a static IP address
It sounds like you want a similar setup (all in 10.0.0.x range) as I have.
- TC: connected to my modem.
- TC: Enabled DHCP server
- Linksys router (running DD-WRT): connected to the TC with a cable running from standard LAN to standard LAN port.
- Linksys router: WAN port disabled
- Linksys router: Wireless is set top bridge

Your theory sounds good. The only thing I can think of is you made a mistake with the configuration. I included some screenshots of my configurations: