I would do the Barklay first. I have a 801 score but had 8 inquiries over the last year and got turned down. 3 of the 8 were from buying a new car, dealership, Ford, and a local bank. 1 other was opening a new checking account. So Barklays can be tough, they said to many inquiries, no US card from us. I do have a Choice CC that I use 4-5 stays a year.
They also said they didn't do a hard pull.