OK, background checks are generally done on what you declare (in terms of your employment experience, academic history, etc.); and something you don't declare (intentionally or unintentionally, like your criminal record, if any, visa status etc.) if you exclude company A from your experience and only declare that your experience is for company B, then company A tenure in effect has no relevance to your current job searching. However, as Ckankonmange, mentioned, if there is a gap due to you not declaring your tenure with company A, then you have to explain the gap; a gap necessarily does not mean something bad, if at all anyone questions you for the gap (which I very much doubt since it is just 21 days), just tell them.
By not putting company A, you are not necessarily hiding it; you feel it is not relevant. However, companies have a last shot question if you have worked for or have any connection to a specific company or any of its competitors etc. etc., if company A falls in the category then mention it if you are asked that question.
I will not say, 'What happens if the BGC agency finds out', I would say 'so what if they find out', you don't feel the company A relevant to the current job searching and you have left it out and have claimed only that you have declared in the CV, in terms of experience and otherwise.