You are not doing anything wrong. Verifly does not work with BA on trips from US to Italy. I have a similar trip next week, BOS to LHR to Venice. I downloaded Verifly and got a little excited when I saw you could add a trip to Italy. But then got the menu you got with only AA and Aer Lingus. For US trips using BA Verifly only works from LHR to US. When I am returning I am going Rome to LHR to BOS, I could not do a trip from Rome I could only do the leg from LHR to BOS. From what I have read there is little time saved and you can just show your documents at check in. I have my seats selected but understand the frustration if you cannot pick your seats because you cannot check in online. Hope that helps.