VFD's can pull up to 200% of their rated output for short periods, if you look at VFD recommendations they may indicate a 50-60A breaker for either a 5 Hp single phase input or a something like a a 5.5kW three phase (HD mode) three phase input run in derated mode. If you are current limited, then adding a DC buss choke can reduce the input circuit protection requirement. But with a 30A circuit, you will be limited as to the power draw and would also dial down the VFD overload parameters. Not an electrician, but NEC code would indicate that the breaker size needs to be a minimum of 125% of the VFD rated input amps, this is not dependent on the motor load. On 3 phase input VFD's run in derated mode, it is model specifc and application as to duty rating and other derating factors. The typical convention these days is to double the VFD size if using it in single phase mode, and that also includes using a DC buss choke. It is unlikely that you would pull a full 5 Hp for any length of time, but I would not undersize the VFD based on that assumption. Also even if you have a 30A circuit, with multiple loads I would be concerned about over heating the wires with any sustained use, household circuits are designed to handle no more than 80% of their breaker rating for sustained periods. This is becoming a failure point for people with EV's and garage charging stations running at close to the rated breaker rating, as well as failure of the plug/sockets.
Typical mid-range VFD's that are rated for a 5 Hp motor would be something like the Yaskawa Yaskawa GA50U
B018ABA 5 HP 230V 1 Phase Heavy Duty Variable Frequency Drive, Hitachi WJ200-075LF (new model is C1-075LFU2) which is 3 phase input run in derated mode, Fuji also has a number of different models. The Yaskawa below, is the least expensive, and are quite good, their software is easier to use then the Hitachi.
Yaskawa GA50UB018ABA 5HP 230V 1 Phase Heavy Duty Variable Frequency Drive The GA500 Industrial AC Microdrive is engineered to help you easily handle nearly any application. Sustainability, flexibility, and ease of use are all designed into the drive, helping you make complicated tasks simple...
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The VFD's are designed to be a direct connection to the lathe motor and not a generic 3 phase power source for the lathe. You also should activate a free run command when applying the manual foot brake and keep the safety interlocks similar to the the original control system design. How this is handled is a bit different for each machine as well as the VFD used.
The BA0018 is a single phase 5 Hp VFD, input current is 35A, so per NEC the minimum would be a 45A breaker. Below is the recommended branch circuit protection ratings for the Yaskawa drives. I typically use a 50A J class time delay fuse for a 5 Hp VFD.