So far, I've not found a connection between Walklet and Walklett. I've seen some mistakes in the spelling of the surname where WalkLET was taken down as WalkLATE (e.g. my great-grandfather Harry Walklet is "Walklate" on the 1881 census, "Walklate" on his marriage certificate of 1898, correctly spelt "Walklet" on the 1901 census, spelt "Walklate" in the Potteries, Newcastle and District Directory of 1907). I've not seen Walklet and Walklett used interchangeably yet, although it wouldn't surprise me if they were used interchangeably at some point.
I had seen an entry for a Thomas WalkleTT on the Ellis Island website (maybe the Thomas you mention??) and thought it might have been the Thomas I was looking for, although I later found him on there as a WalkLET. "My" Thomas (Thomas Albert Walklet) was the brother of my great-grandfather. He emigrated sometime around 1896 and lived at 153 Broadway, Long Branch, New Jersey. He was followed by his brother James Walklet in 1906 and his brother Frederick in 1913 along with his widowed mother Susan.