I would think that most clubs would allow additional training. However at times it's complicated if the training costs are per team instead of per player. The dreaded parent trap then gets going and "why is little Johnnie training with our team, he hasn't paid part of our training fees."
As for the semantics of the "Premier" team. Again some of the parent trap. What defines a premier team? As Chico said it's the most competitive team. As for the parent trap part, why call it Premier if it's playing in the same league with the Challenge teams? Some parents like to see more concrete examples, therefore it's somewhat driven by the league. Regardless, I think every club has what they consider the most competitive team in the age group.
Some clubs really struggle with the multiple teams in an age group. I mean, after all, how could little Johnnie not be on the 1st, A, Purple, Black, Premier, or whatever the club calls the most competitive team. You know that parents are the most objective evaluators of talent when it comes to the age group their child participates in.
The beauty of all this, our society is based on choice. If you don't like your situation, you have the choice to move on to one where you think you will be happy. Of course, some people are never satisfied.