My first answer wasn't as reliable as I'd like. "Ren" could error out in a few build scenarios which are not terribly unlikely. So the following is a bullet-proof fix.

<?xml version="1.0" encoding="utf-8"?>
<Project Sdk="Microsoft.NET.Sdk">
    <PropertyGroup>
        <AssemblyName>Foo</AssemblyName>
    </PropertyGroup>
    <ItemGroup>
        <Content Include="LICENSE" CopyToOutputDirectory="Always" />
    </ItemGroup>

    <Target Name="RmLicense" AfterTargets="Clean" Condition=" '$(OutDir)' != '' ">
        <Delete ContinueOnError="false"
            Files="$(OutDir)\$(AssemblyName).License.txt"
        />
    </Target>
    <Target Name="MvLicense" AfterTargets="Build" Condition=" '$(OutDir)' != '' ">
        <Move ContinueOnError="false"
            SourceFiles="$(OutDir)\LICENSE"
            DestinationFiles="$(OutDir)\$(AssemblyName).License.txt"
        />
    </Target>
</Project>

There are two build tasks, one which executes after a Clean operation, and the other after a Build operation. During the clean operation the renamed output file (if present) is deleted. It is during the build operation that we rename the output file. Note there is a condition. I don't fully understand why, but during the build process there is a time when the $(OutDir) is empty, and no such file will be found. Now we can't just blindly ignore errors (ContinueOnError="true") because during normal operation we want to know if there is a problem here; a missing file or inability to rename the file is an exceptional condition which should correctly break the build. So we skip the operation in the event the $(OutDir) is empty but otherwise perform our work.

I've tried this in multiple projects and it works just fine, so I'm very happy with this. Many thanks to @JonathanDodds for pointing me in the correct direction.