I've had a similar thing happen with a ticket booked through a TA. They, for some reason, cram my first and middle name into the first name field on the ticket. This just prevents you from checking in online. When I got to the airport the agent was able to fix it manually in the system and check me in. This did not seem to affect seat selection on the BA leg, as I could log in (using the BA PNR) with the incorrect name and choose seats without issue (for some reason this did not mess up the linking with my OW status).

In your case, I'm not sure how easy it will be as you are missing part of your name. How did this happen? If it was a TA screwup, I would recommend pushing them to cancel and rebook the ticket and eat the cost as they are the ones who messed it up.