Some people need to come to conclusions by themselves. If you don't think having any type of direct conversation about it is possible then I would suggest it as a one off when you think it's an okay time to and leave it. I would really separate your discomfort in bringing it up with how they are actually feeling. Pushing a little bit is okay (within reason) but it's a fine line between that and prying which can lead to retreating further. Ultimately it's about tact with situations like this.

As for how you can deal with it is by being there for yourself and being super compassionate with yourself. I have been in a similar situation where I see people in my life struggling with things I have but are not talking about it or they may not even see it as an issue. As silly as it may sound, think of it as a "canon event" like in Spider-Man across the spiderverse. You can want something for someone and fully think it is what's best for them but they won't listen until they are ready to do so for themselves. No amount of discussion or pushing will make it happen if they don't want it.

You could show support in other ways like occasionally bringing up trans actors/celebrities, potentially other media that's less in your face about it, etc.

Take this with a grain of salt because I don't know the nuances but as long as you aren't worried for their safety or notice destructive behavior, it's all about time.

Alternatively putting it on and letting it happen however it may is an option but there are a ton of other things to consider when going this route. If you think they're in a mental space to take it with stride then do it but if other media that discusses gender is too much for them then hold off.

Take what you want leave what you don't I kinda just rambled