Rewrite Location Header When Using mod_proxy

Question

We have the following scenario we are trying to proxy requests to two backend domains through the same public address.

<VirtualHost *:443>
        SSLEngine On
        SSLProxyEngine On
        SSLProxyVerify none
        
        ServerName public.mycompany.com

        SSLCertificateFile "/etc/pki/tls/certs/public.mycompany.com.pem"
        SSLCertificateKeyFile "/etc/pki/tls/private/public.mycompany.com.key.pem"        

        ProxyPass /  https://private.backend.com
        ProxyPassReverse /  https://private.backend.com/
        ProxyPassReverseCookieDomain ".private.backend.com" ".mycompany.com"
        ProxyPassReverseCookieDomain ".private.backend.com" ".public.mycompany.com"
        
        ProxyPass /bakend-login  https://login.backend.com
        ProxyPassReverse /bakend-login https://login.backend.com/           
</VirtualHost>

The public domain is (hypothetically for illustrative purposes) public.mycompany.com. The backend domains are login.backend.com and private.backend.com We are using the following mod_proxy directives

The first proxy directives work fine and we can see redirects from the backend to itself being replaced with the public hostname as in

Location: https://private.backend.com/somepath becoming https://public.mycompany.com/somepath

However, if the redirect from the backend is to login.backend.com, we are not able to replace the hostname with the public domain before returning to the client. What we would like to do is replace Location: https://login.backend.com with Location: https://public.mycompany.com/backend-login Is this possible with mod_proxy? We tried using mod_headers at the end of all directives to replace the header, but we were not able to. This is what we tried.

Header edit Location "^https:\/\/login\.backend\.com\/(.*)$" https://public.mycompany.com/bakend-login/$1

We also tried another proxy directive for the redirect, but that doesn’t seem like it would work.

ProxyPass /bakend-login  https://login.backend.com
ProxyPassReverse /bakend-login https://login.backend.com/