Displaying Tweets from CPT

I have a content aggregation website coming online, and have setup a CPT that is being populated with Twitter search results through WP RSS Aggregator.

The permalink for each post is the URL to the tweet. I know WordPress will parse links to tweet and format them for display.

How can I loop these permalinks and have WP process them? Every plugin I’ve tried only pulls from Twitter, and won’t pull from a CPT. I can loop them with php or Custom Content Shortcode plugin, but they don't get parsed by WP.