In the UK the companies register is centralised at national level but the Kamer van Koophandel is not, it is on a local basis. I always translate it as Chamber of Commerce because the Chambers are also organised locally.

... for example, because of space limitation, you could try "company reg. no." or simply "reg. no."

FWIW, I wouldn't leave it as KvK. Surely the point of translation is to render the text comprehensible to a readership that doesn't understand the source language? On the other hand, if this piece of information is irrelevant to the readership - bearing in mind the *purpose* of the translation - then you could always leave it out. Or untranslated if its absence would be too conspicuous.