To my knowledge there is no relationship between the two. They just both share the Black Tiger name. Sil Lum Hak Fu Mun and Fu Jow Pai are two totally different styles. The Black Tiger Clan or Hak Fu Mun Black Tiger sect as you wrote has been a pretty secretive society ever since Black Tiger was created.
The Black Tiger clan has no relationship to either White Tiger or Fu Jow Pai. Hak Fu Mun is a southern Black Tiger style acredited to Soo Hak Fu (Soo Chi Er) one of the Ten Tigers of Canton (Kwangtung Sup Fu) I can't help you in regards to Fu Jow Pai, Bak Fu Pai or Shantung Black Tiger.
jeff
少林黑虎門
Sil Lum Hak Fu Mun
RIP Kuen "Fred" Woo (sifu)