You can set special tax types against products in Brightpearl (read more here), in which case Brightpearl will share the type with Avalara when calculating taxes. However, order rows that aren't linked to products don't have this option. This is true for shipping and miscellaneous order rows.
In the Avalara connector settings, you can specify the AvaTax tax code to use for shipping rows. Brightpearl identifies a shipping row by its nominal code which is specified at Settings > Company > Accounting: Accounts.
Navigate to the Avalara connector settings via the App Store, then specify the tax code to use for these shipping rows. This is usually 'FR' but a full list of AvaTax's freight tax codes can be found here.
It's possible to add a row to a sales order in Brightpearl that is neither linked to product nor that uses the shipping nominal code. These are miscellaneous order rows. They can be attributed to a product tax type by using the 'Product tax type to use for miscellaneous items' field in the connector settings.
Note that neither field is mandatory. If left blank, we use Avalara's default tax type for Tangible Personal Property (TPP): P0000000.