Also, having the ability to logout through that area is fine once the user is in through the MemberPress login form.
hide the wp-login.php page
you can do a redirection from that login form to the other.
Is there any way to turn off the login form portion of the (bbPress) Login Widget though so that it only displays the profile portion when a user logs in?
this is kind of funny since you dont want the login widget for what it does , but i guess you can use a plugin like jetpacks widget visibility and show it only if users are logged in.
there are a bunch of plugins that do what jetpack widget visibility module does.
like this plugin for example
https://wordpress.org/plugins/restrict-widgets/