The client edge can reach the Internet through ISP1 only if downstream routers inside the client network have a usable route toward the Internet. Since ISP1 provides full Internet connectivity and ISP2 is used for direct exchange with selected third parties, the edge design should advertise a default route into the client network.
This gives internal routers a simple, scalable route for all Internet destinations while preserving specific routing where necessary. Running separate VRFs for each ISP may be a valid segmentation technique, but it does not by itself tell downstream routers how to reach the Internet. AS-path prepending toward ISP2 influences inbound path preference from external networks; it does not solve internal default routing. Filtering outbound advertisements so the client advertises only its own originated routes is important to avoid becoming a transit AS, but it is not the mechanism that enables internal users to reach the Internet through ISP1. The clean design is to receive or originate a default route at the edge and advertise that default to the client routing domain. Reference topics: BGP Internet edge design, default route advertisement, transit prevention, ISP peering policy.