This morning, while I was trying to proxy traffic to this site in Burpsuite, I ran across an SSL handshake error. Googling the issue returned this helpful article that got me started on the right path. The crux of the problem was that the JRE didn't have the Java Cryptography Extension (JCE) Unlimited Strength Jurisdiction Policy files installed. However, since this article was published, Portswigger began bundling the JRE with Burpsuite itself.
The solution to fix JCE-related issues (as evidenced in Burpsuite's Alerts tab) was to search for the term “Java Cryptography Extension (JCE) Unlimited Strength Jurisdiction Policy Files 8 Download” on which led to the download page for JRE 8's JCE Unlimited Strength Jurisdiction Policy. You can identify which version of the JRE Burpsuite is using by running:
/path/to/Burp\ Suite\ Professional.app/Contents/PlugIns/jre.bundle/Contents/Home/jre/bin/java -version
Download and unzip the JCE policy files and copy
/path/to/Burp\ Suite\ Professional.app/Contents/PlugIns/jre.bundle/Contents/Home/jre/lib/security/
Start Burpsuite and your problem should be solved. If you've managed to eliminate the JCE error message in Alerts but you still see an SSL handshake error, try resetting Burpsuite's SSL options in User Options > SSL > Java SSL Options. In my case, I had checked “Disable Java SNI extension”. Unchecking this (or selecting the gear icon in the “Java SSL Options” section and clicking “Restore Defaults”) and restarting Burpsuite solved the issue.
Now I can proxy traffic through Burp for sites implementing AWS's TLS certificates.