As some of you know, I think materialism is baloney. I subscribe to the much more parsimonious and skeptical notion that reality is in a trans-personal form of consciousness, of which we are localizations -- like whirlpools in a stream. This ontology is often called monistic idealism ...Any philosophical position, including Idealism, that asserts that reality as we can know it, is fundamentally mental, is accurate to the extent that we have no other way of knowing anything. So your argument is coherent within the context in which it is formed, which is entirely subjective. For convenience we'll call this situation "subjective reality". As a consequence, contrary to what your challenge seems to suggest, the idealist cannot possibly form any refutation for the state existence that applies to things that exist beyond one's mind. We'll call that situation "objective reality".
The best any idealist can do is deny that objective reality exists, and that position wears very thin when we begin asking questions like, "How old is the universe?" and, "When did we humans first appear?" because for those realities to have happened in accordance with Idealism, our minds must have existed before Earth formed and life ( including us ) evolved, and frankly, that idea is just nonsense.
By assuming that consciousness is in biology, as opposed to biology in consciousness -- the latter being, of course, the necessary implication of the idealist position -- you are begging the question of materialism. To an idealist, biology is an image of certain processes in mind, which doesn't deny that there were other, older processes in mind before biology emerged. This is so basic it hurts.
An idealist denies what I call a strongly-objective reality: a reality outside consciousness. But an idealist -- for NOT being a solipsist -- acknowledges what I call a weakly-objective reality: a set of experiences that are shared by multiple human beings and other life forms. Notice that weak-objectivity does not imply strong-objectivity. The latter, as you acknowledged yourself, is a logical inference, not an empirical observation.