As is usual in these things, after hours of searching and ending up posting a question on SO, I found the answer - and in another SO question: How to fetch admin profile details of a company page using Company API - Linkedin
A list of all the companies the user is admin for can be obtained here: https://api.linkedin.com/v1/companies?is-company-admin=true
This is actually mentioned in the API docs but I didn't pay much attention to that page because I assumed it was all dealing with getting information about pages for which you already had the ID. Assumption - still the mother of all f$%& ups.