I don't know about these dojo's being specifically for monks, although I suspect not. I know of a dojo at the Meiji Jingu shrine in Tokyo, Tanaka Shigeo sensei, one of O Sensei's students was the dojo cho, although I'm not sure if he still is. Very fearsome chap but very nice Aikido, my sensei whilst I was living in Tokyo is one of his students. Neither Tanaka sensei nor Nojima sensei are monks as far as I am aware
Whilst you're down in the Kansai area, try the Seibukan Dojo in Kobe, especially asaren (early morning practice) with Nakao Sensei, well worth a visit.