Completely fault-tolerant LTM VIP for DNS requests with reselect to prevent any dropped requests?

Env: F5 LTMs running 11.5.2

We have DNS servers that are exhibiting sporadic issues, timing out in responding to authoritative requests. We also have some DNS clients that are very sensitive to DNS failure behavior (any timeouts cause issues). To address this temporarily (while we figure out the DNS server issue), we're attempting to configure "fault tolerant" LTM VIPs that will detect if a request timed out (e.g., after 1 sec), and re-submit the request to a different back-end real server, if so -- all invisibly to the client.

We've tried setting "Action on service down" and "Reselect tries" at the pool level, but this doesn't accomplish it - the associated processing only kicks in if a monitor marks the server as down. Request queueing doesn't seem relevant (well, maybe for the TCP DNS requests - but those are almost non-existent).

Any guidance on how we might accomplish this? Or get as close as possible, via some combination of maybe a very low health monitor internal/timeout ... any ideas appreciated.