I think he dealt with a possible terrorist in a way anyone would. If she had a gun, he'd be called a hero now. In those seconds you act or you don't. She had forcefully entered where she shouldn't be and should expect to be forcefully removed.
And yet hundreds of people who saw her clearly didn't consider for one moment that she could be a terrorist - why else can you see them in the video watching her quite calmly? Likewise, Field never for one moment thought she was a possible terrorist, for all the reasons stated by IcedPurple. If she'd had a gun, the situation wouldn't have arisen, because she wouldn't get through security.
And she didn't "forcefully" enter. She walked in like everyone else. I'm sure she expected to be removed using reasonable force. She didn't expect to be the subject of a prolonged assault.