The job of HR is to make sure that all of your work-related problems with a legal name change are taken care of, and it is your responsibility to provide them with all that information. You do want to make sure that your money go into the right bank account and that the right Social Security account is being credited. But unless you are on a personal services contract with the company where you are working now, this is really all the responsibility to the consulting company that put you into that job. They are the ones paying you.
The consulting company is responsible for providing such and such service from their employees with certain specified skills. Within the pool of persons with those skills, you are just another slot. As the company where you are working has accepted you from the contracting company as having the right skills, they can hardly complain that just because you are presenting differently in your dress and make-up that you have lost that skill set.
Being on a contract complicates it for you a bit. While the company where you are working can offer you a job as an employee, it cannot fire or discipline you because right now you are not their employee.
So going to HR in the company where your consulting company has sent you to work and fill their contract between the companies can, out of the goodness of their hearts, help smooth the waters for your transition, But you have no right to that help, as you would have if you were to be an employee. It is the HR of the consulting company that you work for that has that responsibility, and the two HR departments should be discussing your transition, and what each can do to make it happen smoothly.
Unfortunately, being a contractor you do not have the rights of an employee, the status for which all the protective legislation is written. In the very worse case scenario, the company where you are working asks the consulting company to bring you back to their offices, and please send someone else instead. And if there were no other work available, this could lead to a very legal layoff. No one has acted in what is defined as discriminatory fashion.
So remembering that as a contractee you have no legal rights, your discussions with HR on both sides should be polite, upbeat and pleasant, not demanding or seemingly litigatious. You want them help you, and they probably will, but noty under threat.
Now why does it all work this way? The companies are not stupid. When pass laws beneficial to employees, especially ones that could lead to costly financial liability, and pretty soon there are many fewer employees and a lot more contractors. Getting a 1099 or a W-2 at the end of the year is a whole lot of difference, even if the money is the same. From the employers viewpoint, another work around is to make you a part owner of the (usually small) company, holding a minuscule share. That also makes you not an employee and therefore not under the protection of anti-discriminatory laws.
In my view, trans should get hired as someone's employee before showing any outward signs of transition, or stating an intention to do so. Your gender presentation of not usually a job requirement. That way the law will be on your side.