It tells a lot when the seller doesn't know what refill the kit takes.
The pictures in the Axminster web site certainly look like 'international standard'. As far as I'm aware, all kit fountain pens take an international standard cartridge. Frankly, I would be very surprised to see a pen kit manufacturer even think about designing a pen around a proprietary cartridge; cartridges are really inexpensive, and it doesn't make any business sense (to me) for a low-volume pen kit manufacturer to invest in developing a proprietary cartridge design and having to set up a manufacturing process given the fairly small number of pen kits that they sell.
So I suggest trying an international standard cartridge or converter.