Welcome to Flyertalk Dubliner1, welcome in particular to the BA forum, it's good to see you here and I hope you can continue to participate here.

I think 2 tags gets you up to 6 legs so it should have been do-able. But on AA they don't through check over 10 or 12 hours (depending on the agent) domestically, and sometimes apply that to international too. There again, if you have a long gap between flights there is some merit in rechecking, since every connection is a risk point.