RE: What do I do with this covering job?
An iron will not do it.
You need a heat gun for covering.
An iron presses things down as is.
A heat gun lets the covering release a bit, so it can be smoothed down with a mitt.
But get the temperature wrong or apply the gun too long and the covering will tear.
Apply it too little and nothing will change... it's a fine line...
What you refer to as "trim covering" is just standard covering.
TopFlite sells a special "trim covering" that is not the same as Ultracoat or Monocoat.
The top flight stuff does not really stretch and contract with heat. It's cut to shape and applied purely as trim. It is sold in strips not rolls and is much thicker than standard covering.