My question is a simple one: is there a group with the properties in the title?
In the absence of the 'finitely presentable' hypothesis, an example is provided by Juschenko--Monod's construction of a finitely generated, infinite, simple, amenable group. On the other hand, all the standard examples that I know of finitely presentable groups without finite quotients seem to be non-amenable.
There are a number of participants on MO who have constructed interesting examples of amenable groups. My guess is that the answer to this question is unknown, but if so I'd also be interested in informed conjectures. Is it at all probable that such a group doesn't exist?