"You need to use a Theme.AppCompat theme (or descendant) with the design library" error

Create a ContextThemeWrapper to wrap the Service's Context with your custom theme, and get the LayoutInflater from that.

ContextThemeWrapper ctx = new ContextThemeWrapper(this, R.style.TranslucentAppTheme);
tooltipContainer = (CoordinatorLayout) LayoutInflater.from(ctx)
    .inflate(R.layout.tooltip_layout, null);

The ContextThemeWrapper modifies the given Context's theme with the one you specify in the constructor. Since a Service doesn't really have a theme, it just tacks yours onto the Service's Context, then the LayoutInflater has the appropriate theme to inflate the design Views.