He just asks a judge he owns, say, an Eileen Cannon, to excuse him because he's busy POTUS....

That's not really how this works. Trial court judges have a lot of discretion, but they don't have unlimited discretion. There are some pretty generic, plain-vanilla rules of civil procedure that apply to every single federal civil trial, and they're pretty well understood.

You don't get to bring a case, introduce evidence on those claims, but then refuse to give the other party a chance to exercise discovery against your client as to the matters raised by that evidence. Colloquially, you can't use privilege as a sword and a shield. You can't raise claims in a lawsuit and then refuse to answer questions about the facts you assert. Anything you refuse to provide discovery on will end up being excluded from the trial.

The courts have made the POTUS "shield" incredibly large, so it's unlikely that he could ever be compelled to testify in a case he didn't want to testify in. But if he chooses to protect himself with that shield, there's little chance that even a friendly judge would let him move forward with his lawsuit. Unlike the CBS lawsuit, where the facts that (ostensibly) supported his lawsuit were unrelated to anything that he could possibly be deposed about, this lawsuit is directly based on Donald Trump's actual business ventures. If he isn't willing to be deposed about his business ventures, he's not going to be able to introduce evidence about his business ventures. No matter how friendly the judge.