Tell me what you guys think  of this and any tips and ideas I could use to flush the lore out a little more 

The main problem you run into is that in the aftermath of the Horus Heresy and the reorganization of the Imperium's military forces, the Adeptus Astartes were separated from the Imperial Navy. Chapters of the Adeptus Astartes wouldn't be chosen, and especially not created, to reunite and mentor the Imperial Navy. At best, an existing Chapter (i.e., 3rd Founding or earlier) might assist in this effort, but the leadership of the effort would fall to the existing remnants of the Imperial Navy.

The last portion, reconnecting lost worlds with the rest of the Imperium of Mankind, is an eminently plausible rationale for the creation of a Chapter in the 4th Founding, however.

The issue with the Red Corsairs is problematic in that it implies that the Silver Consuls ultimately defeated the Red Corsairs, or at least severely weakened their operations, yet the official lore paints a much different picture. I think that you can preserve the core concepts - Heretic Astartes destroyed the Chapters's original homeworld, the Chapter became fleet-based, the Chapter defeated the Heretic Astartes and has developed expertise at fighting such pirates - without name-dropping the Red Corsairs. It might even be better to change it from the Red Corsairs to some other minor Chaos fleet or pirate band, with a homegrown enemy being ideal as that will enable you to develop some opposition that doesn't run into problems with the official lore.

Based on your post, the only lore about your Chapter that I have read, the core that you're looking for is a Chapter that is expert at fleet actions and which has a long history with pirates. There is a lot of room to do that without going over the top or challenging official lore.