Short of raw SQL, can I query for multiple attachment metadata that have a given array key?

While you can do this with WP_Query, I’d do this the way you are trying to avoid– save a list in a separate key. The “pros” strongly outweigh the “cons” in my mind.

1. LIKE queries with wildcards, especially with leading wildcards,
   are not efficient– aka, “slow”
2. and a LIKE query on serialized data is going to be prone to error
3. You will likely be running this query on display, which means that users see a delay

Whereas…

1. The “space” needed to store another value in, say, the options table
   is negligible
2. And the query to retrieve the value is as about as efficient as a
   query can be
3. The work is pushed almost entirely to the backend, when the data is
   saved, so that any delay is seen by the administrators/authors of
   the site and not by the ordinary users.

You can add and retrieve the data with the core functions update_option and get_option, the latter of which has caching mechanisms built in.