F5 LTM Virtual Edition at 2 data centre

For the replication of F5 BigIP config from PROD to DR, create a SyncOnly Device Group. This should work perfectly fine, regardless of distance to another participants of Device Group.

I would not configure LTM network fail-over across 10km WAN link. You will risk having your PROD and DR active at the same time. A manual intervention may be your most reliable option to switch between the sites.

Not sure what your data-center setup is, but if you have external routers that maintain a full BGP routing table, that can also contribute to reliability. Re-announcement of your external IP-addresses can occur automatically and no human intervention would be needed in case of ISP problems. In all other cases, going with the manual human intervention should work the best.