Not fake as much as machine-made, i.e. still a katana in all essence, but not handcrafted. See, most katana you get from webpages, curiosity stores and whatnot are machine-produced, often made with cheap materials and meant for ornamental purposes only. While technically actual katanas they don't have the cutting power or the strength, and are often brittle (since they are made with low-carbon stainless steel; this is more brittle than traditional high-carbon steel), and the blade tends to be attached poorly.
Now, the katana you buy in a handicraft store in Japan, looking away from tourist-intended replicas, are handmade, high-carbon steel, they are so sharp you could cut yourself from looking at them, so blank you could put on makeup and style your hair staring into one, and so durable it could cut steel.
£3000 I would put in the lower tier of high-quality katana, definitely approved.
I cosplayed Emperor Lelouch in April last year. I've also cosplayed Link (from Legend of Zelda), and Darth Vader (borrowed a friend's costume). Nowadays I just use my yukata+zori, and if I bother, my shinai
yukata = light summer kimono
zori = sandals (not the wooden kind with heels)
shinai = bamboo sword used in kendo (modern sword art)