I used to collect canes from different places I'd travel to, and the idea you have sounds really good; the problem is anything more than just a basic design would be pretty thick, and likely to be fragile.
I had a cane that was from Aruba that was shaped like a lizard, with the handle being the lizards head, but the carvings were pretty much as basic as could be, and still the thing was fragile.
TBH I'm not sure how much you're wanting to spend, but like climbtree said you'd probably want it made from some metal. Wood would probably snap, especially if it had chunks carved out of it. I had one that had copper branches around it, and it seemed to work but iirc it cost like 30-40 dollars more than the average cane.
Coincedentally, I do have a wood burning pen, and I'm not sure how it would look on a round cane, but what you're talking about is definitely doable as far as burning.
I would probably look up businesses on the internet and try and get quotes.