Which carrier/itinerary would you pick? The first 3 segments are what i care about in term of bags going though and IRROPS issues. I'm planning to use a TA to book it all under one PNR.

RDU 5:25pm-JFK7:16pm on AA connecting to
JFK 9:30pm-LHR 9:35am on BA (T5)connecting to
LHR 11:45am-LCA 6:20pm on BA (from T5)

(more flights later in the trip on AA mostly but they aren't as relevant as there are no connections).

or

RDU 6:15pm-JFK 8:09pm on DL
JFK 9:30-LHR 9:40am(arrives T3) DL coded but operated by VA
LHR 11:45am-LCA 6:20pm(leaves T5) on BA

I'm mainly worried about the chance that our bags will not go through from Virgin/Delta to BA in Heathrow

I'm also looking at
RDU-JFK on DL
JFK-Ath on DL
ATH-LCA on Aegean (again not sure about having to get bags out in Athens)

I'm mainly worried because i'll be traveling alone with 2 young kids (husband will join me there a few weeks later) and have heard horrible things about BA's IRROPS procedure - I have heard better things about AA but I have experienced major delays and a cancelation with them in the recent past. I have heard great things about Delta, I understand they are better about intelining bags to other airlines (under same PNR) and love the ability to track bags and rebook in cases of IRROPS right from their app.
Thanks for any input!