The dyed versions of sophisticated storage barrels and chests do not have recipes for the create mod. Factory gauges cannot be used to automate these containers, either through mechanical crafting or shapeless crafting.

As seen from JEI, the only recipe possible is the normal crafting table, while base versions of these containers have the automatically detected create recipes.

Playing in 1.20.1 with:

• Create 6.0.5 for mc1.20.1
• Sophisticated Storage-1.20.1-1.3.49.1167
• Sophisticated Core-1.20.1-1.2.69.1005
• Sophisticated Storage-1.20.1-1.3.49.1167
• Sophisticated Storage Create Integration-1.20.1-0.1.12.36
• Sophisticated Backpacks Create Integration-1.20.1-0.1.3.11
• Sophisticated Storage in Motion-1.20.1-010.19.104
  as well as wealth of other mods that probably aren't relevant. Happy to provide a more thorough list if requested.
  Thanks for the awesome mods!

Based on what I am seeing in code, this should still be craftable prooperly with mechanical crafter, it just requires extra addition in jei to appear in the list of one of its recipes (similar to how I have to add it for JEI to show it as a regular recipe as well)