Why are Democrats pushing so hard for an FBI investigation into Kavanaugh given that he already had 6 background checks?

Because it's a two-fer for Democrats:

1) They get to delay Kavanaugh's confirmation and possibly completely negate it.

2) They get to kick and screm impeachment for Trump should he in any way try to intervene in the investigation, even if it drags on and on and on, delaying filling the SC seat. Firing RR would then be called interfering with the Mueller investigation and the Kavanaugh investigation and fuel their call for impeachment.