What is the best way to simulate a Click with MouseUp & MouseDown events or otherwise?

I would use a Button control and overwrite the Button.Template to just show the content directly.

<ControlTemplate x:Key="ContentOnlyTemplate" TargetType="{x:Type Button}">
    <ContentPresenter />
</ControlTemplate>

<Button Template="{StaticResource ContentOnlyTemplate}">
    <Label Content="Test"/>
</Button>