The burger chain said it had received "unexpected" responses from other brands and had already began developing a pop up shop for sale of the burger, which it proposes should be created using a "signature" ingredient from each restaurant.
"Let's make a burger merger", the brand said in the letter. "All we need from you guys is a simple 'yes."
Addressing McDonald's it added: "Our original proposal still stands. McDonald's, please take your time. We're totally at peace with that."
Last week, McDonald's MD Steve Easterbrook said on the brand's Facebook page:
"We love the intention but think our two brands could do something bigger to make a difference. We commit to raise awareness worldwide, perhaps you’ll join us in a meaningful global effort?
"And every day, let's acknowledge that between us there is simply a friendly business competition and certainly not the unequalled circumstances of the real pain and suffering of war.
"We’ll be in touch. Steve, McDonald’s CEO. P.S. A simple phone call will do next time."
(This article first appeared on MarketingMagazine.co.uk)