Two Considerations for a Solution
There are two considerations when offering a solution to the Problem of Evil from a Presuppositional standpoint. In order to look at the first, let's reconsider the formalized statement of the problem:
a) God is all powerful
b) God is all loving
c) God knows that evil exists
d) Evil does exist
e) Therefore, God does not exist
Notice that this takes the form of a deductive argument, meaning that the conclusion "God does not exist" follows with necessity just as long as every one of the premises is true, and just as long as the form of the argument itself is valid (which in this case, it is).
An important aspect of the argument above to recognize is the difference between premises a-c, and premise d. In presenting this argument, the unbeliever is stating premises a-c as hypotheticals; that is, according to the claims of Christianity, premises a-c are true. However, premise d is presented as fact. 9 times out of 10 the unbeliever does not say "the Bible claims that evil exists", but rather "evil exists". Let's consider why this distinction matters.
When presenting the argument above, the unbeliever is attempting to demonstrate the truth of all of his/her premises in order to demonstrate the conclusion is true. We, as believers, "give" premises a-c to the unbeliever, since those are the attributes of God found in the Bible. However, we don't need to "give" premise d to the unbeliever, at least not at the outset.
Please note that I am not claiming that we, as Christians, do not believe evil exists - we obviously do. However, as the burden is on the unbeliever to make their case, there is a real benefit in pressing them to demonstrate that premise d is true. After all, if a demonstration of the existence of evil is not forthcoming from the unbeliever, then they have failed to prove that God does not exist. The purpose in taking this tack is not to avoid the question of whether evil exists, but rather to press the unbeliever's worldview to demonstrate their inability to make the argument in the first place.
With that said, ask the unbeliever to demonstrate that evil exists. In doing so, be sure to clarify that they must offer an objective demonstration of this, if they wish to demonstrate their conclusion that God does not exist. This is exactly the point at which you can press the issue of worldviews, as the unbeliever does not have an objective, non-arbitrary set of presuppositions to use as a foundation to demonstrate that evil exists.
Short of offering an objective foundation for the existence of evil, the best the unbeliever can do is turn to you and state "but don't you believe evil exists?" This is exactly what you want them to ask!
We will cover the second consideration next time around, and offer the Biblical answer to the problem.