Any free DNS benchmark tool for comparing a local server with public ones?

There is an open-source project called DNSBenchmark that was made for this kind of comparison. It may be worth trying if the paid tools you've found don't have the features you need.

The version described in the discussion supported IPv4 only, so check the current project details if you need IPv6 testing.

Before picking a tool, decide whether IPv6 matters for your tests. An IPv4-only benchmark can still compare local and public servers over IPv4, but it won't tell you how they perform over IPv6.