2nd Edition is very picky about which races get which classes and how far they can advance. This is to do with the game’s method of power balance.
In the original Advanced D&D rule set, Multi-Classing and Dual classing are funky/complicated things. Only demi-humans can Multi-Class and even then they have maximum level caps for advancement in each class that are specific for each race.
Next up is Dual Classing. Dual classing is a human exclusive — whereby you start your adventuring career over at level 1, re-level, and once you are back to original level, gain the benefits of each class. But, you never again advance in the original class.
So in 2nd edition class availability (and thus game balance) is heavily dependent on your race. 2nd Edition believes in factoring in the lifespans of the various races into its calculations of game balance. If not, elves (whom had the longest lifespans) would have the most powerful warriors and wizards throughout every realm where they existed, since if you have a 1000 years to gain experience, you can then max every class level (or reach ridiculous levels) in comparison to the shorter lived dwarfs or humans.
So the compromise is to allow demi-humans to split their XP amongst multiple classes simultaneously, and allow them to advance only so far in particular classes. Some races can advance further in some classes than others. Classes that are particularly powerful or quirky (such as paladin) are labeled a Human exclusive. Humans can only advance in a single class at a time (Dual classing is possible). This means that being human awards you the strongest single class advancement alongside access to some of the most powerful classes, though at the expense of the racial bonuses everyone else gets.
By limiting the transmuter to Human and Half-Elf, they are limiting the mechanical multi-classing / dual classing access of the more powerful and quirky class.
Yes, talk to your DM about your character Sheet.
Your DM is also your best source of information and best ally for ensuring your character is both what you want and what works with the table. Perhaps your DM won't mind letting you "have" you racial/class trait combo, or perhaps it might be a smidge outta whack and a few minor adjustments may be needed. Letting him know what you have lets him know what kind of adventures he can expose the party to.
Ensuring that each party member doesn't greatly overpower or fall too far behind the other characters is really important for keeping AD&D fun and engaging for everyone at the table.
That, and they will eventually find out anyway. D&D isn't you vs. the DM, it's everyone at the table vs. the adventure. Working with the 5th party member (the DM) to make things clear and understood to ensure mechanics and story are acceptable goes a long way towards making great games and adventures.