You have several different laws to account for. As I understand the law, you would not be committing a Penal Code violation by having the weapon in your car, in your safe, or on your person in a child care center that you own. The Penal Code says that it is legal to carry your weapon on your own premises or premises you control. Assuming you are a sole proprietorship, it is your premises. Assuming you incorporated or filed it as a partnership, it is still premises under your control. This means that a police officer will not arrest you for carrying the weapon there.
There is a big caveat here. There are some strict laws on keeping the gun safe from children. Your plan looks like it would meet the law.
Then there is the DFPS rules. They say you cannot have the weapon on the premises. This is a law, part of the Texas Administrative Code. The bad news is that there is no definition I can find for premises in that law. The definition accepted by the courts, that I am aware of, would make the whole property - including the parking lot - the premises. This is NOT the premises used in chapter 46 where there is a very specific and narrow definition. I base this on the way the courts have looked at licensed premises for drinking. Your plan would clearly violate this law, even if you left the gun in the car. The good news is that violating the Administrative Code is usually a civil violation. This means that the worst DFPS could do is file for a fine against you or revoke your license to operate a child care center.
I do not recommend breaking the law and placing your business or wallet at risk. But that is a choice for you to make.In addition to the law, consider who would know about the gun and how you would get caught. If you don't ever need it, a gun left in your car should never cause a problem while you are at work (no one would know and no one would complain. There is always a chance of getting caught by someone you did not expect to see when you are carrying the weapon inside.