I would inquire if your employer would be willing to help you get your license. If they are willing, take them up on it. Businesses have no qualms with firing you over lunch on Friday with no notice, so I have no problem using their benefits for my own advantage.
However, no, it’s bad form to steal customers from your current employer after you leave. If they come to you and ask you for services that’s a different story, but you cannot approach them morally. (You can do so ethically, if you honestly believe you will provide a better service for them than your current employer. Utilitarianism says greatest good for the greatest number.)
I think of tuition reimbursement, medical, etcetera as payment for services rendered. You have rendered services and they are offering to pay you with contingencies - the contingency is to get your license in this situation.