Unless there has been a change in the last few months, the SP centre at the Mitchell Library does not have the facility to save any documents to a USB/Flashdrive.
Sorry if I gave you the wrong info Denise - I've never been to the Mitchell Library.
I have been to the Burns Monument Centre in Kilmarnock where access to SP can be purchased for £15 a day and they did allow USB/Memory stick to be used. But it seems Mitchell Library don't

I would advise you to contact the Mitchell Library by email and let them know when your trip is /what you are looking for and ask them when you should book your place. Access to Scotlands People will let you search birth/death/marriage/census and wills. But it won't have Carnegie Orphanage records .
I don't know where they can be found but I think chrissey's suggestion to ask The Church of Scotland is a good one.
Looby