Plenty of barstools will work; the problem is in attaching any sort of tray. I know; I've tried!
What's required is a barstool that's largely wood; then you have to add 'L' arms to the sides and base. The 'plastic bucket' you found at Ikea won't work because there's no easy way to attach arms, and without arms, there's no support for a tray.
Once you've made and attached the 'L' arms, the top of the 'L' needs hardware that will allow a tray to slide on and off when you're sitting in the highchair. The easiest thing to do is 'rob' hardware from an existing highchair.
The real problem is the tray. Most plastic 'baby' highchair trays won't work because you have to move the screws outbound past the sides to accommodate your barstool's extra width; you're effectively stuck having a wooden tray made. If you have woodworking tools or are otherwise handy with wood, this may be a project you'll want to tackle yourself. The three or four wood craftsmen I've approached regarding my request with raised eyebrows. I told the last one I wanted the tray for a highchair I was making for an elderly relative. His reply: "Yeah, right." He told me he had other projects lined up and wasn't interested in helping me. I don't have the time or tools to do it myself, or I'd try.
Good luck!
