Can I use a VIP to route HTTP traffic through a proxy?

You can achieve this by using the "Forwarding (IP)" virtual server type. In the VS you can limit the accepted source to the server IP, and to limit the destination you can either create one VS per port or a unique VS and associate an iRule that will check for the destination port.

In this configuration the F5 will act like a router and will look at his routing table to determine the next hop (probably the default GW). In the iRule, You can explicitely specify a given IP as your next hop for your selected ports, and use the default GW for the other ports. See the nexthop command.

This is a pure routing approach. You can instead use a full proxy approach using as many "Standard" VS as your selected ports plus one catch all Forwarding (IP) VS to forward the rest of the traffic (or one VS + iRule for port filtering). In this approach you can leverage the standard VS advantages for your selected ports.