How do I order authentication options correctly in Apache, by location?

The order required is:

1. Basic Auth - IF Authorization is in the header
2. SAML Auth
3. For 2 specific locations, no auth

This is what I have, but it does not seem to work correctly

<Location "/">
   AuthType Basic
   AuthBasicProvider ldap
   ...

   <If "-z req('Authorization')">
      AuthType Mellon
      MellonEnable auth
      MellonEndpointPath /mellon/
      ...
   </If>
</Location>

<LocationMatch "^/(notifyCommit|shuttingDown)/">
   AuthType None
   Allow from all
   Satisfy any
   Require all granted
</LocationMatch>

However, any time I try to access either of the two locations where authentication should not be required (notifyCommit or shuttingDown), I get a 401.

What am I doing wrong here? I am under the impression that Location and LocationMatch are merged together, and the order is top-to-bottom. If this is true, it should evaluate "/" first and see that auth is needed, but then find the location match section, and see that auth is not needed for /notifyCommit/ or /shuttingDown/.

I've gone through a few other related questions and tried a bunch of options, but none seemed to work, including moving the location match before the location section, and SetEnvFf Request_URI ....

EDIT

1. For the two locations where authentication is not required, no username or password should be needed. It is basically anonymous access.
2. The SAML/Basic auth conditional section seems to be working fine.