Request that it be changed, especially since there will not be a last name on your BP for the security check point. I could tell you how to reprint it with a good name, but some people frown on manipulating a computer printed bp.
If you OLCI, and this is domestic, the only people that are going to care about the name on the BP are the people checking ID at the checkpoint.

However, AFAIK, it is NOT a requirement to provide ID to travel domestically.
You would have to go through secondary screening, but I believe it is possible. On an early morning SJC-SNA flight, I gave my ID to the GA to get a new BP, and forgot to get my ID back. On my return I had to go through secondary screening.

Edit: I just thought of something. If you can't get the AA to change it, perhaps you can talk the TA into giving you a security pass with your correct name.
These are the kinds of passes you get when picking an UM, for example. You can also get them for going to AC when not flying. One advantage of booking through AA.com is reservations booked while logged will default to your registered name first. (A hazard when booking for someone else, if you forget to change it)