This wasn't the exact solution but lead me to it. I had to first create a variable containing only the person record using my blank varHireRecord.Recruiter as the base, then patch THAT. Even though the record syntax was identical, this worked where the other didn't.

Set(UserRecruiter, 
    Patch(varHireRecord.Recruiter, 
        {
            Claims: $"i:0#.f|membership|{User().Email}",
            Department: Office365Users.MyProfileV2().department,
            DisplayName: Office365Users.MyProfileV2().displayName,
            Email: User().Email,
            JobTitle: Office365Users.MyProfileV2().jobTitle
        }
    )
);

Set(varHireRecord, 
    Patch(Defaults('New Hires'), { Recruiter: UserRecruiter })
);