The reason for the vapor barrier is that any moisture in the air inside (includung your exhaled breath) will migrate out of the building partly though the wall unless there is a airtight barrier. When the warm moist air reaches its dewpoint the moisture will condense. Here it is cold in the winter(sometimes also in the spring and fall and occasionally in the summer). The real world result of this is that the air migrating though the insulation(and it will without a vapor barrier) will reach its dewpoint somewhere in your insulation and the moisture will condense there. Eventually you will have soggy insulation which will lower the R factor and maybe worse promote rot in a wooden frame and in your case rust in your building frame.
My dislike of chicken wire is related to my early experience with it and not relevant to your situation. That aside I still would not use it in the fashion you indicated. You are going to lack the vapor barrier and the insulation is going to get contaminated with smoke, dust, grinding particles and who knows what else. Much of that is likely to lower the R value and render your walls almost impossible to clean except by removing the insulation and replacing it with new stuff.
There are probably fire ratings and cautions online for various types of insulation which you should check and take a small piece of it and put a torch to it to see if you can get it to burn as a low tech test. Do so in a well ventilated area or better still a fume hood as some types of insulation give off poisonous gas when on fire.
Sounds like a nice project by the way,