I'll give you the email, but I do not think you'll need to use it when you read the email I got back to them when I had your question.
permissionrequests (at) ea dot com
Here's the email I got back when I asked
Greetings:
This is in response to your request dated February 28, 2013. Unfortunately, EA is not able to provide a blanket permission to use our copyrighted game footage. We have previously advised YouTube that EA generally does not object to fair uses of video footage or screen shots from publicly released versions of our copyrighted games, including on YouTube channels which are commercialized, but that we reserve rights to address inappropriate use of our copyrighted content. We will continue to advise YouTube directly of that position.
Thank you,
Permission Requests Team
Electronic Arts Inc.
In other words, the answer is, Yes*. *=unless they find it inapropriate.
There exist hack videos, trolling videos with BF3/EA's content. Yet they don't take them down. But I do know they take down alpha/beta footage when the alpha/beta isn't over. So gameplay such as yours listed, including the fact they're monetized/comercialized, I'm certain you won't have a problem. As long as you don't upload alpha/beta footage, I don't see them filing a DMCA against your videos.