I’ve considered speaking with our department head about it privately, but I’m an adult and not tattle, so I think I should talk to A about it directly.

NTA There's tattling over trivialities to polish the teacher's apple and get off on a sense of power over others (which is the type of tattling most people hate) and then then there's the the type of "tattling" which is actually protecting yourself and others from people committing hurtful, selfish, even criminal acts. This has been labelled "tattling" by these self-serving, often abusive people precisely to give it a stigma so that victims won't do it -- avoiding this type of "tattling" only benefits those who abuse others for their own benefit.

Your friend repaid your help and kindness by utterly letting you down and giving you a bad attitude to boot. There's no point covering for them considering they are making no effort to work properly or give you any sort of consideration at all. There's no point talking to her either, considering that she is taking this attitude from the outset. It's a choice, not an accidental misstep.

I would argue that this won't really reflect badly on you if you speak to your department head with your concerns. Frame it as you coming to them because you want to take responsibility for being the one to recommend her. You had never seen this side of her before and you don't know if it is something new or something that has only become visible to you because you are now working directly with her; regardless, you feel that as you were the one to vouch for her, you should step up now to speak once it has become obvious that the way she is at work is nothing like you had presented to the company. You don't really like coming forward like this, but to remain silent would be to become complicit.