Yes you have a shot. I know Yale has recently been actively recruiting top STEM talents like yourself, so I would look in to applying there: Inside the Hahn Scholars program’s push to recruit top STEM students - Yale Daily News
How and why did Chick-fil-A hire your before legal age? Personally, I would mention the job as an EC but not draw attention to the illegality of the situation unless there’s some special story there.
For Caltech, look here: Research Papers and Publications - Undergraduate Admissions - the LoR from the site of research they ask for might need to be more narrowly focused on the details of the research and your contributions than the general letter written by your BUGSS mentor, so you may need an extra letter or for your mentor to modify it for this specific case.