Since BPD is a primarily emotional condition that often has its sufferers unable to face shame or blame, or responsibility for their own feelings and actions, telling them they have a mental illness can be very hurtful to them, and counterproductive for you.
They spend so much time doing mental gymnastics to make things other people's faults, being told "you have a mental illness that causes all this pain," is really no different than, "it's pretty much just your fault so go get some help already."
I find the terms "depression" and "anxiety" and "poor self-worth" to be safer to use. H does not deny he has the first two, and has trouble when he tries to deny the last one. And, he can also have all of these anyway in ADDITION to BPD, as these things are not mutually exclusive.
Just like any problem, if you can take care of parts of it, sometimes the other parts are more manageable. I may not be able to wash ALL my dishes, but if I can get the big stuff washed so it can be put away, the smaller things are more manageable. I may not be able to mow, weed eat, trim tree, paint the house, all the same day, but I can tackle one or the other and that makes the others less tiresome to get around to.
If a pwBPD is dysregulating about travel or anything else in life, you can validate their feelings by asking about anxiety they feel and try to have a discussion on that end.
The term BPD help us a lot more than it helps them, in my opinion. It seems to take a lot to get someone with BPD into T and for T to actually make headway. In the meantime, we now have ways to look up tools to assist us in forming better relations and drama mitigation tools.

You can treat someone who swims, waddles, and quacks like a duck without telling them you think they are a duck - the actions are more important than the labels.