VIP control across data centers - how to ensure only 1 VIP is up at a time?

for the ospf/bgp integration, you can use Route Health Injection
https://my.f5.com/manage/s/article/K15923612

in f5 dc1, create a vserver and add dc2 syslog vserver as the pool member.
add above new dc1 vserver as negative/reversed health monitor target of syslog dc2 pool.

you can use these either 1 of these 2 monitor config to create negative/reversed health monitor.