Before you try to fool SnapPea, remember that you'll almost certainly have to go above 16 (17?) crossings to do so - see https://doi.org/10.1007/BF03025227 for the tale of the tabulation of knots by Hoste and Weeks and, independently, Thistlethwaite. Here is a nice quote: "...our methods for nonalternating knots are not algorithmic. Instead we simply employ a collection of methods that *work* for $N \leq 16$."

Edit: Ok, I googled "complicated unknot" and found a paper "Hard Unknots and Collapsing Tangles" by Louis H. Kauffman and Sofia Lambropoulou and a thesis "Interactive Topological Drawing" by Robert Glenn Scharein. I went through both and entered the unknots they give into SnapPea. In all cases SnapPea says that the volume is zero and, futhermore, reports that the fundamental group is $\mathbb{Z}$ (one generator, no relators). The unknots they discuss include the Goeritz unknot, Freedman's unknot, and several unknots that require increasing the complexity of the diagram before decreasing. (ie via Reidemeister moves). Another knot that SnapPea handled (~55 crossings) was the one on page 135 of the thesis, which is claimed to defeat KnotPlot.

SnapPea would report the results so quickly that I will conjecture that Newton's method, hyperbolic geometry, etc were not really involved. Instead, I think that SnapPea's retriangulation heuristic "detected" all of these unknots. That is: SnapPea takes the diagram you give it and produces a straightforward triangulation which is linear in terms of the crossing number. It then cleans this triangulation up, getting rid of material vertices and doing 4-1 and 3-2 moves wherever possible. I think that this first step must be getting rid of almost all of the tetrahedra.