I think Rosebrae Nursery (Garden) used to occupy that site
I remember a Rosebrae nursey on Pensby road next to Sandy lane but that was in Heswall.
It is now block of apartments/flats built on the land.
I can see the Pensby hotel from my lounge window but don't know of any earlier History
other than it was built about the same time and style as the Arrow park hotel.