Flaps on a bag should be no problem, but like printing on a pocket or near something that will interfere with the print stroke, it helps to build up the platen.

This block is what I use for a chest print, it wouldn't be big enough for a larger print like the bag you showed.

My buildup is in there, holding the printable surface above a pocket, those strings, zippers, seams - who knows?
You may have to get creative with wood and duct tape. Some of my best work on troublesome surfaces were the ugliest collection of odds and ends.
Your ink choice is based on the material.
For leather, or as a last resort I use the excellent T & W 11000 series of polyurethane ink. It sticks to anything. Glass, leather, rocks, concrete and of course synthetics.
TW Graphics Solvent Base Ink
It air dries, so that means it DRIES IN THE SCREEN .... and will not come out. Cleans up with MEK.