When clicking an invoice link from the SCMA overview page or invoices page, I get a permissions error. When clicking the invoice link from the transactions history page, I can see the invoice. I can't change the permissions on a copy of the customer center role. Has anyone dealt with this before?

Yes. The invoice page makes an additional call to OrderHistory.Service.ss (requires Sales Order Permissions), while the transactions history page does not.

Isn't it possible to include a child view on the account overview page e.g. the balance data from the balance page; or something that would allow the data to be available at that level?