You need to import the logger:

import logging
_logger = logging.getLogger(__name__)

Then something like this:

        # add logging
        _logger.info('------------------------')
        stock_moves = self.mapped('move_lines')
        for stock_move in stock_moves:
            _logger.info(stock_move.product_id.name)
        _logger.info('------------------------')
        #

I added these lines to the start of the action_assign method of the stock.picking model, which runs when you click CHECK AVAILABILITY.

Output for Chic/OUT/00002 using the demo database: