What is this "DNSLookupDuration3s" you talk about? Is it an alerting rule? Can you show the expr?
To me, it sounds like the opposite problem. My guess is that blackbox_exporter is first making a UDP DNS query, and either the query or the response is being blocked. So after 3 seconds it retries with TCP, and that succeeds.
You can check this theory using tcpdump (especially if you can do tcpdump on the caching resolver as well). Do you see an outbound UDP DNS query, but no response? The resolution is to fix the underlying UDP communication problem.
Are there any virtual machines involved in this? That's the one case where I have seen this exact problem before with UDP traffic but not TCP. The packet is sent without a correct UDP checksum, because checksum offloading is enabled and the client expects the NIC to insert a correct one; but the receiver doesn't know this, and just sees a packet with a bad checksum and discards it.
The solution, or at least workaround, is to disable UDP transmit checksum offloading on the VM's network interface (probably just the one running blackbox_exporter)
Try:
ethtool --offload eth0 tx off
and if that doesn't work, also try:
ethtool --offload eth0 gso off gro off tso off