You should put down your former name, but I wouldn't draw attention to having done it personally. Not putting a former name down could look not like you're lying per se, but withholding truth, which can look just as bad. Some change their name for less justified reasons to try and hide criminal pasts. If you get red flagged for that it could just draw more attention to you.
Best of luck with the offer/job! I'm sure that if you've been given the offer that you've sold yourself enough to the company that they won't let something like this retract their offer. And if they were to do so, then they're not somewhere worth working for anyway! Also, anything HR discovers about you can only be shared if relevant to the job. That would be things like them discovering a criminal past and you're supposed to be handling money. Or you have mobility issues yet set to work hard manual labour. Anything else they can't, or at least shouldn't share unless relevant to position, which of course being trans isn't! I'm starting work at a place I worked at before changing my name and yet they're still sticking to this rule. Of course people I worked with before are going to know, but anyone new will only find out from either myself or other employees, HR cannot tell them.