I hope this doesn't come across as harsh, but....
He's the DM. He can do what he wants.
The rulebooks are only guidelines. The DM is the one, powerful creator of the world you game in. If he doesn't think the idea of gnomish bards or dwarven wizards fits, that's his perogative. The rulebooks do not have the final say. The DM does.
Personally, I'm all for that. I don't allow dwarven wizards, nor do I allow halfling wizards (I'm not quite certain about gnomish bards). I WILL allow orcish paladins, however. (And, I don't believe in level limits..) If a player doesn't like it, he can leave. There are many more players that will be willing to play with those rules.
To quote the Afterword of the 1st edition DMG:
"It is the spirit of the game, not the letter of the rules, which is important. Never hold to the letter written, nor allow some barracks room lawyer to force quotations from the rule book upon you, if it goes against the obvious intent of the game. As you hew the line with respect to conformity to major systems and uniformity of play in general, also be certain the game is mastered by you and not by your players. Within the broad parameters given in the Advanced Dungeons & Dragons Volumes, you are creator and final arbiter. By ordering things as they should be, the game as a whole first, your campaign next, and your participants thereafter, you will be playing Advanced Dungeons and Dragons as it was meant to be. May you find as much pleasure in doing so as the rest of us do!"