I'm skeptical about the idea that a Lannister showed up, made it clear they were trying to ensure the loyalty of this house, then said "by the way your kid is a mage, we're taking him now" and the guy just let them. None of the possibilities are terribly nice, especially considering the broadsheets he should have read by now.
Can we do a divination to check if the kid is bound yet? Vague accusations of mage binding are one thing, proving Tywin enslaved the second son of a crownlander lord is another, and would probably flip the whole region.
The real trick would be to rescue him while disguised as other people; perhaps concerned Alchemist remnants, Old Gods worshippers concerned about druids being taken, or something like that? "Independent" confirmation of one of our worst accusations would implicitly support all of the rest of them.
Now that I'm thinking about it, if Tywin is comfortable pulling this so quickly with one noble kid, why would he only do it once? He might have handfuls of slave mages from the aristocracy we could save. Any region he pulled this on would instantly see this as an existential threat, and it makes for a good lead into exposing his bound lords.