My advice would be:
- First, don't do hyphenated last names when those hyphenated names are nothing more than spelling variations. Pick one variation, and document the others in the Biography. Because the hyphenated name is not a true last name, it is only an artificial name that should be removed.
- You can also use the Other Last Names field to list possibly used spelling variations. You can have multiple names in this field, separate each name by a comma.
- How to pick a spelling variation? Ideally if you have documentation on the subject.
- What if there are multiple spellings in the documentation? See which one is used more in all of the documentation, OR see how the person's descendants are spelling it.
- What if there is little (to no) documentation on a subject? Use the most common spelling variation found in the person's descendants.
- It's better to pick some spelling variation than to leave the profile in some weird limbo, waiting to decide. If more, or better, information comes to light in the future, then the profile can be changed.
Those are my rules of thumb. However, others may have better advice.
Regarding this specific situation, it looks like the descendants of profile Piersol-Pearsall-4 (found here) use the spelling "Peirsol". This can be seen by picking a random descendant, such as Harry Allen Peirsol, who has a Find A Grave page, which shows a photo of his gravestone that clearly has that spelling variation.
So if I were you, I would pick "Peirsol", and be done with it. Note the possible spelling variations in the Profile's Biography section, even making subsection about it if you desire.
Again, my two cents.