PhD interview - travel arrangement I'm sorry if this is a wrong place to ask this question but I hoped to hear from others who travelled long distance for an interview.

Finally after countless applications I was invited for an interview next month. I'm trying to arrange my journey there (London to Glasgow). I'm trying to keep the expenses to minimum. Fastest option is a flight BUT I would need to leave home from 3/4am to make it on time (interview at 10am, home to airport via coach, then flight, then airport to institute via bus). That is if there are absolutely no delays.

My alternative would be to take a coach (8-9h), stay overnight in a hostel. And then the next day meet them all fresh and ready.

So I emailed them politely asking them if this arrangement would be acceptable stating that I worry I might not make it on time if I travel on the day of the interview.

They replied to me stating 'no' and that my statement is wrong (about not being on time) and went on to say that there are plenty of flights on that day. They sounded very upset at me.

So now I made a wrong impression even before meeting them. I even made them search for me and made them wrongly accuse me of trying to visit Glasgow at their expenses.

I get a very bad feeling about all this and I still would rather not go with the plane option. It's not only about being late but also arriving tired and dishevelled (formal suit).

Am I wrong to feel a bit angry at them? I would keep the travel expenses to minimum and it would be a huge difference not to travel overnight with multiple changes.
Take the coach, stay overnight and pay your own expenses; forget this flap ever happened; do your best in the interview.  Keep in mind that the office staff who arranges for travel is probably overworked, and has NOTHING to do with hiring decisions, and will have almost nothing to do with you once you are working there.  Focus on figuring out if the job is a good fit for you -- read up about the department or whatever before you go, think about how you can make a contribution to their efforts, and how you will convince them of that.  And enjoy your trip.  Even if this is not the place you end up working, the interview will be valuable experience for you in your job search.