Thanks for the feedback. Keep on playing around.
I plan to surface the font, palette, etc info for each logo just as you've described. Specifically so that you can use it to drive design/branding decision further in your product/company/site.
For now (and seeing that you're a "code jedi") I'd recommend just scoping out the X-Logo and X-Meta HTTP headers for the logos you've saved, eg:
$ curl --head -s "https://api.logocaster.com/v1/logos/new.png?action=recreate&name=companyco&style=bubble&palette_id=1643&font_id=438&bubble_width=18" | grep "X-"
X-Meta-Permalink: https://api.logocaster.com/v1/logos/new.png?action=recreate&name=companyco&style=bubble&palette_id=1643&font_id=438&bubble_width=18.0
X-Logo-Name: companyco (edit)
X-Logo-Style: bubble
X-Logo-Palette-Id: 1643
X-Logo-Font-Id: 438
X-Logo-Bubble-Width: 18.0
X-Logo-Drop-Spaces: false
X-Logo-Text-Transform: none
X-Logo-Font-Category: sans-serif
X-Meta-Palette: #72ab1a #2aade4 #9ed4d6
X-Meta-Palette-Background: #fdfcfc
X-Meta-Font-Family: Montserrat