With respect, I suggest that Luitgarde of Vermandois is unlikely to have been Bethel's/Bathel's mother. The research that I've done on Ancestry.com (for what that's worth) and Geni.com (more accurate perhaps) suggests that Bethel was born in 870, while Luitgarde was born in 920.
I have been following my Palmer lineage backwards (through my mother) and thus have arrived at Bethel/Bathel of Centreville. I can't find his parents but it seems clear that he was of viking stock--either a companion of Rollo, or already in Normandy via earlier viking raids and settlement.