I've just spoken to the customer again briefly. We're having a whiteboard session next week with their AWS provider to thrash out a design.
From the link Vladimir Yakovlev shared before, I think the closest to our proposed design is the last diagram:

AWS will essentially be the core/hub. None of the branch sites need to communicate with each other, but they will all have Direct Connect up to AWS, and they will have one firewall in AWS that all sites go through to get to the Internet.
Apparently in December last year AWS introduced a new transit VPC which now allowed sites to route through AWS on their way to another destination (i.e. the Internet), so maybe that will be good enough to do what we need it to? I'll find out more next week...
Matt