Edited to say:

Can all these segments be booked on AA 001 ticket stock as one PNR? My understanding is that the AA IRROPS policy is much better than BA and AA is better about interlining to their partners.

6/13 UA 3444 RDU 5:40pm-EWR 7:29pm
6/13 BA 188 EWR 9:30pm-LHR 9:20am (on same day)
6/14 BA 662 LHR 11:45 - LCA 6:20pm
7/11 Cobalt (new Cyprus airline - flight sometimes marketed as APG 386)
LCA 11:00am-CDG 2:45pm (doesn’t show up on all search engines -I might have to book this segment separately)
7/14 BA 1535 CDG 11am-JFK 1:35pm (actually AA45 but cheaper if booked as BA)
7/16 AA 4622 LGA 2:35pm-RDU 4:29pm

Thanks!...

Can all these segments be booked on AA 001 ticket stock as one PNR? Does anyone have a TA recommendation that can book it to ensure our bags can go through on the RDU--JFK-LHR-LCA leg of the trip? This prices out significantly lower if priced separately that all together on AA or BA.com. (1 adult and 2 children flying this exact itinerary in economy, my husband will have a different travel date on RDU-LCA leg)
Numbers 1,2 etc indicate roundtrip options:

1. 6/13 AA 4344 RDU 5:25pm-JFK 7:16pm
2. 6/13 BA 114 JFK 9:30pm-LHR 9:35am (T5)
3.6/14 BA 662 LHR 11:45 (T5) - LCA 6:20pm

4. 7/11 Cobalt (new Cyprus airline - flight sometimes marketed as APG 386)
LCA 11:00am-CDG 2:45pm (doesn’t show up on all search engines -might have to book this segment separately)

2. 7/14 BA 1535 (AA45) CDG 11am-JFK 1:35pm (cheaper if booked as BA)
1. 7/16 AA 4622 LGA 2:35pm-RDU 4:29pm

Thanks!