In my opinion the best repros are available from http://www.lawranceordnance.com/
They do have the velcro to allow adjustment of the headband (hi Mike) but they are very well made.
To bring a repro closer to authenticity one would have to provide ventilation between the headband and the helmet shell. Ventilation being of paramount importance with sun helmets.
One would have to remove the existing headband and make 8 cork spacers, as shown below, to be glued onto the helmet shell and to the headband. The other method used was of corrugated buckram or leather between the shell and the headband but a much more difficult task.
The original spacers may have been riveted to the shell and I notice on the one repro that I have the headband is secured by pins which have two prongs pushed through holes in the shell and the headband liner and then bent over to secure the headband. Theoretically these could be used for both the spacers and the headband.
A good repro will have hooks for the chinstrap rather than having the chinstrap fixed permanently to the helmet.