The motor you linked to is single phase so wouldn't work with an inverter.

If you are going to buy a new motor and inverter I strongly suggest that you contact Newton Tesla and get their advice on which of their motor/VFD packages would best suit your needs. Though I did my own install of a 3ph motor and inverter on my mill, when it came to my Super7 in the end I just bought their package and it worked without issue.

Or fit one of the modern brushless servo motors/controllers made for industrial sewing machines. Look up "Jack Sewing Machine Motor" on eBay. Quite a few people have used these and recommend them for mills & lathes.