When you implement a button (or tool) you have to define its 'id' in the config.daml file. Usually the item template for buttons/tools will define an id for you. It looks like this:
<tool id="CompReporter_AddCompBySelection" caption="AddCompBySelection"
className="AddCompBySelection" loadOnClick="true" ...>
<tooltip ... </tooltip>
</tool>
the 'id' in this case is: CompReporter_AddCompBySelection
In the 'code behind' for your dockpane you can then use your tool (or button) just like any other ArcGIS Pro button (built-in or not) by using this snippet:
// use CompReporter_AddCompBySelection
IPlugInWrapper wrapper = FrameworkApplication.GetPlugInWrapper("CompReporter_AddCompBySelection");
if (wrapper is ICommand toolCmd)
{
// if it is a tool, execute will set current tool
toolCmd.Execute(null);
}