There is no better way to get a cat to leave a memento other than attracting them visiting your yard multiples times.
In short, attract the cat you want to your yard as many times as you can, it is actually not hard when you already have all the information about their favourite food/ toys. Just make sure you have them available on all time. You can check this post here for more information about Neko Atsume mementos.
From most of my cases, it usually takes me at least 10 times for the cat to leave me a memento. I believe there is an engagement scores where their system will reward you a memento after a certain period of revisit.
So please make sure you attract the cat as many times as you can, that way, in my opinion, is the best way to get the cat to leave you memento(s) as soon possible.