Yes.

Create a new Field on the mrp.production (Production Order) model to store the finished product lot:

Add this field to the User Interface via your own custom View:

This pushes the value of the custom field to the Produce wizard and sets the default company and product values for the newly created Lot.

<xpath expr="//button[@name='open_produce_product'][1]" position="attributes">
  <attribute name="context">{'default_finished_lot_id': x_finished_lot_id}</attribute>
</xpath>

<xpath expr="//button[@name='open_produce_product'][2]" position="attributes">
  <attribute name="context">{'default_finished_lot_id': x_finished_lot_id}</attribute>
</xpath>

<field name="bom_id" position="before">
  <field name="x_finished_lot_id" context="{'default_company_id': company_id, 'default_product_id': product_id}"/>
</field>

You can then assign the Lot before you start production, or even during the creation of the Manufacturing Order:

When you click PRODUCE the Wizard will be pre-populated with the Lot from the Manufacturing Order: