I don't know where you live, but here in Canada the importance of having the legal name is for two reasons, and the primary reason is to do a background check. If you're not giving them your legal name, which they need in order to do the background check, then they'll likely find out and not hire you/take back the job offer. The other thing is for pay check reasons, though that isn't as important as the background check/making sure you are who you say you are, you don't have a criminal record, haven't had cases of gross misconduct at jobs in the past etc.
On the other hand, it's not like they are filling out the backcheck form for you based on the resume, so if you put your chosen name on the resume, you just have to be sure to put the legal name as the primary name on the backcheck form, and just explain to them at that point that your legal name is X, but you prefer to use Y.
You'd also have to make yourself aware of whether or not the programme/company/place where you live has laws to protect trans people from discrimination in the work place, or that the company has a functioning diversity programme in place. Either way, you'll have to at least come out to your employer, though neither you nor the employer need tell other employees if you want to remain stealth.
In all these types of situations, I always tend to think its a good idea to talk to an lgbt councilor who's aware of the legal aspects/what your rights are, before proceeding and making assumptions that might get you into trouble where you live.
And of course, you could circumvent all of this by changing your name legally
