I think you do have to say something, but the best way to say it is not, "I am trans, is that a problem?" as you seem aware.
I would phrase it as a question about their administrative policy in reviewing your application. You can say, "I am transitioning female to male" or "I have transitioned female to male" (depending on how far in the process you are), "and I want to make sure there is no confusion with my application, because my name/resume/etc. is male, but my records and references (degree, job, etc.) are all in my female name. How should I handle my online application to make it easier for you to see/review/gather my materials and assess my candidacy? I don't want anyone to be inconvenienced or confused."
Then throw it to them. The answer could range from: GAH! A trans person! Go away! to Oh, someone did this two years ago, and we didn't have a problem with it to the more likely, we don't have a policy about this but people have changed names before, so I'm sure we can handle it if you note it x way (or i'll get back to you or don't worry about it...) etc.
Yes, it requires you to out yourself to them. But that makes sense if you will be going there and they will know anyway.
In general, though, I think the best way to do it is make it about the administrative/paperwork bureacracy aspect. The fact that you are trans is no big deal. Just that it might make your application non-standard from the typical application they see and you want to conform to whatever streamlined policies they have or get an individual recommendation from them about how to handle it so your application can more easily be processed by them. End of story.
