I am also a nursing mama of a 2 year old (29 months), with no plans to stop anytime soon! I'm also a breast cancer survivor, diagnosed at age 26, ten years ago, in remission since then.
I had a chest x-ray a month ago and researched it ahead of time -- they are totally safe for nursing. The radiation does not stay in your body.
Breast MRIs and mammograms are different -- there are varying opinions about whether they're appropriate/effective/safe for nursing moms, so I haven't had either since I got pregnant. But lucky for you, that's not what you need.
Based on what you've written, it seems perfectly appropriate (and probably important) to have the x-ray they recommended.
I know it's harder to do when you're feeling vulnerable and scared, but ignore the stupid nurse who clearly has not been educated about the health issues related to breastfeeding. File her in the same place one files ignorant relatives and obnoxious strangers: in the Ignore file. Hopefully you won't cross paths with her again as you go through this. If she works in the office where you'll need to go, it's probably possible to request a different nurse -- quietly tell the person at the front desk that you had a bit of personality conflict and if there's a way to request a different nurse, you'd really appreciate it.
Sending you lots of healing energy!