Hi Max, The best place to start is to look at the boiling points of flavor active compounds in the hops and then compare to those of compounds that you are more familiar with. Additionally, the flavor threshold of each compound needs to be considered.
Common Compounds (not comprehensive)
Humulone (Alpha Acid): BP 488 C - Very low volatility, unlikely to have significant impact
Myrcene (Essential Oil): BP 64 C - Boiling point is just below that of methanol, expect this to congregate at the front of the run
Humulene (Essential Oil): BP 99 C - Boiling point is just below water, so expect this later in the run
Caryophellene (Essential Oil): BP 129 C - Boiling point is a bit above butanol and within the range of the amyl alcohols, so expect this predominately at the back of the run.
As a general rule, any flavors from bittering hops that persist through the boil are not volatile enough to make it into your spirit through distillation. The reason for late hop additions is to preserve the volatile aroma compounds. Most of these would be volatile enough to make it into the spirit depending on your methods and cut points. Thus, aroma hops would likely have a greater influence, though this would have lot to do with the specific hops selected. I would also recommend against boiling your wort with the hops in order to preserve the aromatics for distillation. Dry hopping your distillate would certainly have an impact. Among other things, it would allow you to extract the non-volatile compounds. Note that hop bitterness comes from isomerized alpha acid, not the "raw" alpha acids found in hops. The later are isomerized by heat during the boil. If you want to achieve this bitterness, you would need to find a way to isomerize the AA or just add an IAA extract.
On a side note, I have tried quite a few hopped spirits, and the only one that I found enjoyable was Anchor's White Christmas (2012), which this post has inspired to fetch from my shelf and have a glass.