Paidnice will wait to apply a late fee until after your grace period ends, but what if it ends during a weekend and the invoice hasn't been marked as paid yet, but it has been paid?

Paidnice has a setting for that - weekday only grace period:

When enabled, Paidnice only counts weekdays in your timezone toward the grace period, so you won't apply late fees on the weekends when there's nobody reconciling payments.