Aditya,
As Youssef is alluding, you're most likely going to have to proxify the request to the other VIPs, that is, pass the traffic through the first VIP to the second. A 307 redirect will cause the client to maintain the original POST method, but there's NO WAY to also pass an authorization header through the client into the next request. So you have maybe two options here:
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Pool the traffic through VIP1 to VIP2 on the other F5, in which case you don't have to send a redirect and don't have to worry about the POST method changing. And you can insert the static authorization header here.
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Have VIP2 insert the static credentials. Assuming the authentication isn't actually being consumed by VIP2, but rather the application servers behind it, it would be conceivable to simply have VIP2 insert the authorization header on the way to the application.