LSP-Omnisharp provides IntelliSense features only.
Build process for complex projects is controlled by
dotnet executable these days (.NET 6+), which also works for .NET 4.x targets.
The most tricky or cumbersome part is just to create/write proper
*.csproj project files files with all the settings and dependencies in it and maybe
*.sln a solution file. Once those exist
dotnet build <solutionfile> is all you need to build your project.
VS Code also just executes the following to compile a project:
dotnet build <solutionfile> /property:GenerateFullPaths=true /consoleloggerparameters:NoSummary