BGP Peering for HA should change the weighting of the BigIP self-IPs on failover, so the upstream router has a preference for the Active device self-IP.
AskF5 Home / Knowledge Center / BIG-IP LTM / BIG-IP TMOS: Routing Administration / Dynamic Routing
https://support.f5.com/kb/en-us/products/big-ip_ltm/manuals/product/tmos-routing-administration-13-1-0/11.html#guid-622a8e11-3e85-45fc-b76a-09b50304d5c3
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About dynamic routing for redundant system configurations
If the BIG-IP system that you are configuring for dynamic routing is part of a redundant system configuration, you should consider these factors:
You must configure the dynamic routing protocols on each member of the device group.
For protocols that include the router ID attribute, you should verify that each member of the device group has a unique router ID.
When you configure Route Health Injection (RHI), only active device group members advertise routes to virtual addresses.
Special considerations for BGP4, RIP, and IS-IS
For the BGP, RIP, RIPng, and IS-IS protocols, you no longer need to specifically configure these protocols to function in
active-standby configurations. Each member of the device group automatically advertises the first floating self IP address
of the same IP subnet as the next hop for all advertised routes. This applies to both IPv4 and IPv6 addresses.
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Make sure your Virtual Addresses are set to enable Route Advertisement
Alternatively, you can use imish to create a specific custom route map to advertise via the floating self-IP.