For a black hole the size of our solar system, could you avoid its singularity?

For a non-rotating black hole...no. The internal geometry prevents avoiding the central singularity for a time much longer than the light travel time from the event horizon to its center...a few hours.

For rotating black holes, in principle you could avoid the singularity by simply choosing not to enter the equatorial plane where the ring singularity lives. Some mathematical studies of these weird objects seem to offer an interesting and non-fatal journey for the traveler, but the problem is that real black holes are very messy. You would probably be killed by the gas and radiation from the infalling matter you are trying to out-run to enter the black hole.