You are going to have a Mission Impossible unless you and your debate partner agree on at least one thing: you must agree to communicate honestly. That means, not to argue out of rudeness, contempt, the desire to confuse with meaninglessness, intentional, or unintentional, misunderstanding(s), failure to agree on definitions, bad intentions against one another, etc.
Your debate partner has already violated this rule: the Easter bunny argument is a meaningless and specious argument thrown at you to muddy up your thought processes. Either that, or, they are seriously muddied thinkers themselves.
There are a relatively small number of unimpeachable “proofs” for God’s existence. You might want to read St. Thomas’s Summa Teologica and get thoroughly familiar with his five proofs. St. Augustine also provides proofs.
Henri Bergson provides an interesting proof from an epistomological perspective. Bergson says that “God is that than which nothing greater can be conceived.” I like this proof because, when one thinks it through, one discovers that he/she can and does in fact conceive of God, in this way.
So, it is not that difficult to conceive of that than which nothing greater can be conceived. But, if we think of God as not having existence, then it is we that have the problem: because, without “existence,” something greater can yet be conceived and we can conceive it and can know it.
The atheist understands what God is, then, denies His existence. This is his undoing.
First, learn and understand all of Aquinas’ five proofs, then learn those presented by St. Augustine, then – as a last resort, if you are sure your opponent communicates with honesty – provide him/her with the fully thought out Bergsonian argument.
God bless and good luck!