I graduated with a degree in mechanical engineering, and the problem you've got are the joints between the 2X3's. In order for the beam to be strong, you need to carry stresses from one piece of lumber to the next all along the length of the beam. Joints between the pieces of lumber make that a problem and complicate the design and sacrifice portability.

Can you have a 12 foot 2X6 ripped into two pieces, each similar in size to a 2X3? That would eliminate joints between pieces of lumber along the length of the beam.