I understand that using an umbrella on Shabbat that was opened prior to Shabbat is prohibited by the Poskim due to Marit Ayin. Can a person therefore use on Shabbat a "Shabbat-Umbrella" that always remains open and never close to avoid any Marit Ayin issues?

The umbrella hat once existed i.e. a hat with a permanent umbrella fixed to it however THAT IS NOT ACCEPTABLE. So as to avoid forming an option of Chilul Shabbat. See also Shemirat Shabbat Kehilchata 24:15 and the footnotes. Anyhow a Shabbat umbrella would not to my mind solve the problem of Marit Ayin. esw