I would approach understanding your question first by taking an imaginary perspective from your son's point of view. When you enrolled him in this "learning center," he was only two years old--surely too young for any learning curriculum and still young enough, in many places around the world, to be expecting his nourishment to come from Mommy's breast--an excellent source of pleasure, social interaction, and nutrition, by the way, the three basic elements of any good meal. There might have been quite a gap between his actual developmental level and the philosophy of the "learning center." I get a sinking sensation--and I imagine that your 2 year old son might have, also--when I saw that "they do not allow you to bring in OUTSIDE food." If Mommy's food is "outside" food, what kind of message is that giving the child?
It is not clear to me whether this "learning center" is really a play group for an hour or so, or actually a day-care agency, but I wonder whether the place has a certain amount of ideological pretentiousness about it, trying to appeal to parents' popular anxieties about nutrition and education and taking an aggressive, rigid, opinionated approach to their convictions. I would guess that your 2 year old was shrewd enough to see through this propaganda, and didn't want any. A small child has a very limited repertoire of responses to things he likes and doesn't like, and refusal to eat is one of the few ways that a toddler can say, "No thanks, pal!" And eventually, your son succeeded in making the whole system cave in anyway, because he got his way in the end: now Mommy is bringing in her food for him, after all. Of course, by now he probably has found aspects of the place that he enjoys, but he has fallen into a rut with his refusal to entirely swallow that alternative, opinionated Mommy. He is still loyal to his own good, nutritious, nurturing Mommy and all the wonderful things that she shares with him from her heart.
Like all toddlers, your 3 year old is torn between his nostalgia for infancy and his fascination with joining the grown-up world. His behavior can be characterized by "back-and forthiness." Food fussiness and food craziness is very common in childhood--and individual temperament probably plays a big role in the stubbornness and finickiness which many children bring to this domain. When a 3 year old feels pushed too fast, he digs in his heels; the stance of opposition can then become a habit, and take on a life of its own. I believe that this may be the situation here.
What to do? I agree with your pediatrician! We need to look at the actual needs of a 3 year old, rather than a lot of slogans. A 3 year old needs to feel large-and-in-charge, and needs to enjoy his own choices of the good food that Mommy and other people who love him will provide to tempt his palate. If the pediatrician is not worried about weight loss or growth failure, I would put your emphasis on making mealtimes enjoyable, delicious social occasions. Naturally, you would not offer a diet of only french fries to your son day after day--but you can safely include everything that ordinary people ordinarily eat.
Branching out and trying new things will happen on its own, once your son feels as though HE is in the driver's seat with his own mouth and his own body. First, he needs to consolidate his sense of security that he is in control, and not being pushed around by the agenda of others. I would put the focus on meal-time fun and human interaction, and pass up a helping of ideology from the food Nazi's. Take inspiration from your persistent little son, and just say no!