What a difficult situation for you, Mumof2. I feel for you.
My reply assumes the immediate crisis is over, and your DD is safe, and you are now 'just' trying to work out what to do about her trip. If this isn't the case, none of the following applies...
There is (IMO) absolutely no way you can say to the host family "Oh, BTW, my DD took an overdose last week, so keep an extra-careful eye on her please"... It is, as you say, an unacceptable burden.
So it seems to me you need to know or find out for certain whether your DD was making an attention-seeking gesture she won't ever repeat, or whether she might. If you think she might - and even a small suspicion would be too much for me - then I don't think she can go on her trip: you can't give another family that worry; you can't expect or trust them to watch her constantly; and their lives as well as yours would be turned upside-down if your DD did it again.
I think you need to talk to her. You need to let her know your dilemma. You need to ask her how she feels about her trip - does she want to go? Does she think it's safe? If I were you, I'd spell it out to her: this is very serious, and you can't let her go if there is any risk at all she might do it again, for all the reasons above. Her responses should help you decide...
Definitely talk to your GP and ask their advice. I have to say, if you or they feel she is a suicide risk, or you're concerned enough to seek a CAMHS referral or other psychological support for your DD, then that probably answers your question for you: you can't let her go. :(