Lavendar a child under age 14yrs old cannot travel alone on long-distance public transport (where only a child ticket is bought one way from A to B). I agree with your suggestion but it isn't a practical option.
Okay, you have done the "I understand you feel sad and bored". Friend's Mum seems to think it is YOUR problem.
Plan B:
Sit down with both girls tomorrow and praise your DD for joining in and looking after her friend. Then TELL DD's friend that you are disappointed/sad that she has not joined in so far and has found everything boring. So..... she can choose what you do today and then tomorrow DD will choose.
(Make sure your options are only things that you are happy to do with the girls - and keep arcades off the list)
Perhaps DD's friend would like to buy a garden game e.g. Swingball, croquet, skittles from her £300 spending money. Alternatively, she may like to buy a beach game e.g. frizbee, beach ball, kites
Once the shopping trip is done - then it can be 'her game'
I know it sounds like I am suggesting over-indulgence, but I am truly trying to suggest ways for you and your family to get out of this mess.
Perhaps DD's friend hasn't been taught how to play. It doesn't come naturally to children. It takes a great Mum/Dad, child-minder, nanny or grandparent to teach children how to play and socialise.
My final bit of advice is to find the strong voice within yourself. Do NOT negotiate with a child who is not responding to holiday opportunities. She will rule the roost! So, how would you deal with DD if she was acting up? You are loca parentis so use your strong parent voice to say, "We don't behave like that in this family. I expect you to ...." (or something similar).
I am rooting for you OP. You offered a kindness thinking that your DD would get more pleasure by having a friend with her. It's gone a bit pear-shaped, but it is resolvable.