I don’t think this property is available in the design wizard, you have to add it via the code page. Also, it only impacts the text space inside the tooltip, not the whole tooltip itself, so it may not be sufficient for your needs.
There’s no way, that I’m aware of that you can add a stylesheet property that would be able to impact the tooltips because of the way the leaflet library renders them, so that tooltip style might be the best you can do.
You should have been able to apply a stylesheet to a map/plan page, however this is not possible in the current UI code. I have submitted a PR to enable this (#3815). Once this is merged, you will be able to add a stylesheet to format the tooltip for a leaflet page: