This practice is very common in Beijing and honestly, until I clicked on the link above, I hadn't for a moment thought it had caught on in dear old Blighty. Count me as one of the people that thinks it's not the done thing.
I have no problem with "bring something special" (e.g. a unique wine, or something homemade, and to share with the chef, with whom one is friends) if one has a longstanding relationship with an establishment. I've seen this done at several world-class restaurants. Never seen it done or done it myself at somewhere where I'm dining for the first time. But actually, the OP wants 'a birthday cake' -- the chef of a decent restaurant can do that.
Anyway, the matter appears resolved thankfully. Hope the birthday celebrations go well!
tb