Now that you have access to the inside part of the spring-loaded battery posts, you can easily push them all the way in. From the inside of the battery compartment, scrape all of brittle white reside off of the metal posts.

You may also need to scrape the brittle white reside off of the other ends of these posts that are outside the battery compartment. Carefully spread the mouse "jaws," taking care not to stress the ribbon cable. Scrape the posts completely clean.

Your goal is to get the spring-loaded posts moving completely freely. If they still hesitate, work them back and forth a few times, then check to see i more of the brittle white reside needs to be scraped off. Repeat this step until the posts move freely.