The simplest solution is to purchase an AC motor drive (VFD or ASD) with a single-phase input or with a three-phase input that is capable of operating with a single-phase input. The same vendor you found sell the same single-phase input drive separately for less than half the money. Vevor is not one of the major brands like Allen Bradley, Hitachi, Fuji, Schneider, Mitsubishi, ABB, Teco, Toshiba, Siemens etc. so it might not be as reliable but will probably be good enough. I think a lot of these off-brand drives are made for cost sensitive applications like HVAC and water well pumping. Many of them are likely to have been made in the same factory with different face plates and stickers. There are only 4 manufacturers of the power transistors (IGBT’s) used in all drives and my experience with all of them has been uniformly good. The off-brand drives will cut corners on the other components (leading to a shorter life) and product support.
The second simplest solution is to purchase a drive with a three-phase input with a current rating that is at least 1.73 X the rating of your current router. One of the three input connections would be left unconnected. It will run at up to 57 % of rated output current. Automation Direct sell some appropriate 10 hp units but they are more expensive and still are also not a major brand. Surplus Center has this Hitachi drive for about the same money.
I do not recommend rotary phase converters. They require using a three-phase motor with a higher rating than your load motor (so it will be larger and more expensive), you will not get proper three-phase out of it and the efficiency is low.
If router you currently have is also rated at 4 kW (5.4 hp), you will need a drive rated for 9.3 hp. The closest would be a 10 hp drive.
The input current through each input loop (1) of a drive with a single-phase input is 1.73 X (square root of 3) the input current through each input loop (3) of a drive with a three-phase input. The input rectifiers will be selected to withstand the rated input current.
Edited to add:
Make sure you install the drive away from the router if you are going to cut metal. Preferably, higher up on a wall. Many have control panels that can be mounted separately from the drive. None of these drives are dust proof. Getting metal dust into them WILL cause them to release magic smoke. Someone on a different internet forum who did not heed this advice reported seeing a bright flash before their drive died.