Hello,
If I right click on a panel, choose panel options, choose add widget, and then double click on LCD Weather Station, then the widget appears behind the round "Panel Tool Box" button on the right of the panel.

It is hidden behind that and unclickable. I see a partial button starting with "co", so I think it is a "configure" button for the LCD Weather Station.

How can I get this widget in a location where I can actually see and click it?

Thanks.

Instead of clicking add, try dragging it from the Add Applets area to the location you wish to place it and releasing there.