I think you’re panicking - unless you have reason to think a hard search would show up large / new debts (are you committing massive credit fraud?!).
Anyone looking carefully at your credit record is going to see the difference between someone in dire financial distress, and someone who switched credit cards / got a new mobile contract in the same month.
If you’re genuinely concerned about the hard searches, consider that you might be better off getting them both done now. How do you know if a lender looks at the number of searches, or the time since you last had one? (You can’t, they won’t tell you).
Edit: also you’ll kick yourself if AmEx take away the 5% introductory cashback offers between now and some arbitrary date which you guess will magically make you look like a responsible borrower.