What do you mean by "radio"? A receiver only, or a transceiver? What are you using it for?

SSB basically quadruples your power output, compared to double sideband with carrier. That's why most hams use SSB exclusively. I'm pretty sure most commercial operations that rely on HF (aka Short Wave) for long distance communications also use SSB rigs.

If all you want is a receiver to pick up BBC World Service or Voice of America, then probably you want (need?) to have the DSB with carrier.

As to your question "would you prefer a 1.3lb radio with SSB or a 0.37LB radio without?" it sounds like you have specific makes and models in mind. In which case I'd want to know (a) what are you planning to use it for? and (b) what are some of the other specs?