The Kurt or the Asian copies are low profile if you remove the swivel base. I bought my 5" genuine Kurt without the base in 1975 and use it on my Rockwell mill, which is a little bigger than a Clausing. I have never wished I had the base. I use an Asian 4" copy of a Kurt on my Hardinge Mill, rather than my original Hardinge 4" vise, because I like the Kurt design better. I left the swivel base, which was included, on this vise because there is plenty of vertical room for it, but it is easy to use the Kurt design vises without a base. A CNC mill makes a swivel base especially unnecessary. I also have a 3" Asian copy of a Kurt, but have never needed to use it. It is not much lower than the 4" size. These import copies cost less than $100 and mine seem to be made well enough, but that does not mean they are all good.
Larry