Hate to say it, but absent some serious abuse, it would be difficult to justify a walk off the job strike that could potentially jeopardize patients. Those things don't seem to qualify. An informational leafleting would probably accomplish everything you are hoping for without making it seem that you are abandoning patients.
More importantly, all this needs to be done in the context of a collective bargaining setup that specifically authorizes such a strike. Without that, any wildcat strike is likely to result in sanctions against the "strikers" and the union as well if the union encourages such a work action. Your union needs to discuss this with a lawyer ASAP. You may be considered public safety workers and have specific cooling offer idols that are required before striking as we'll.
Likewise, as noted while there are whistleblower protections and protections against retaliation for legal collective bargaining actions, in no circumstance can that guarantee you that your educational evaluations won't be affected in some way for fellowship or job prospects. Most doctors are so skittish about interpersonal problems at work, so it is hard to believe that there aren't some places that won't want to take someone that might cause a problem, no matter how justified your cause it.
My suggestion is for the union to find a way back to the table. A 3rd party mediation may be of help. The hospital wants to resolve this as well. I can't seem them blocking things that might improve care for little or no cost. The salary item is really the toughest part. You may need professional help to demonstrate why your demands are reasonable, etc and to understand their position to find a way to come to a compromise.