Your "bridge" AP will offer no additional protection for any traffic going to/from the Internet. However, depending on how you have your network configured, it may be able to provide protection for local traffic between your own devices.
As it stands, your path to the Internet is this:
You --> Open WiFi --> Internet
You're proposing some sort of configuration that changes the path to this:
You --> Secure Bridge --> Open WiFi --> Internet
As you can see, your traffic is still passing through the Open (and insecure) WiFi network. Traffic flowing between your devices and the Secure Bridge will be encrypted and therefore protected from eavesdropping. But any traffic that is internet-bound will have to be sent across the Open WiFi network in the clear* - otherwise the Open WiFi network wouldn't know how to handle it.
*Note: Here, "in the clear" is referring only to the physical layer of the transmission. Your traffic may be protected by encryption at higher layers (e.g.: HTTPS) in certain applications.
The only protection added is for local traffic sent between your own devices which are connected to your Secure Bridge. That is, traffic passing between your devices that does not need to be routed through the Internet (e.g.: file transfers via network share). Since this traffic is not Internet-bound, it will never need to be sent over the Open WiFi and therefore will be protected by the encryption your Secure Bridge uses even if it is not encrypted at a higher layer.
The only way to keep your traffic secure when there is an untrusted network in your path is to make sure it is all encrypted at a higher layer. Typically, this is done via VPN tunneling (e.g.: Tor). VPN tunneling effectively changes your network path to this:
You --> VPN Provider --> Internet
Of course you still need an existing network, such as the Open Wifi, to connect. But, with all your traffic passing through a VPN, nobody on that network will be able to casually eavesdrop on your transmissions. You also need to bear in mind that the VPN provider will have just as much access to your traffic as if it was being sent over an Open WiFi network, as will anyone along the path from the VPN provider to the Internet endpoint - endpoint included.
For any data you want truly protected in transit, you need end-to-end encryption at the highest layer possible. This is usually done via application-level protocols like HTTPS, SSH, and PGP.