You are correct. "Destructive Device" has no barrel length definition, thus you are free to shorten the barrel to any length you wish with no ATF notification or approval requirement.
However, you are incorrect in your assumption that you need to keep the original barrel somehow. There is no requirement to maintain the original barrel length in any way. You may cut that 18" to any length you wish and that's the end of it. You may notify ATF of the change in barrel length, but it is not necessary.
Rather than change the barrel, it is far easy and cheaper to simply cut what you have. You will not easily find a spare Streetsweeper barrel for sale. and installing it is not easy.
One bit of caution in all this, you should confirm that your state agrees with the federal definition of "Destructive Device" here. In cutting the barrel, you may be creating a "Short-Barrel shotgun" under state law while being perfectly compliant under federal.