How long to wait before contacting another arm's length referee?
You are very considerate to your referees, and I hope you get this postdoc position!
There is no exact number of days for these things and you would know the person's emailing habits better than me. Perhaps in another 2 days (4 days total), you can write him something like:

Dear Prof. XXXX,
I am just writing to follow-up on my email below.
Of course there is no rush for your assistance with refereeing my application package, because it is not due until 13 August. I wrote to you this early because I wanted to make sure that you had plenty of time, since I wanted to inconvenience you as little as possible.
If I may, I would just like to ask whether or not you might be able to act as one of my arms-distance referees for this application package: The deadline is not until August 31st, but I would just like to first establish who is going to have enough time to do it and who doesn't, so that if for example you are not able to do it, I don't end up asking someone else with not much time available. The idea is for me to be as considerate as possible to the time and energy spent by my external referees.
Sincerely,
user125706

Please feel free to adjust this letter, or write your own based on it, or copy-paste it. Whatever makes you feel most comfortable.
Good luck on getting this position!