An employee with an H1-B is definitionally not an illegal immigrant. H1-Bs are definitionally documented legal workers, the companies using undocumented workers are not companies interested in getting visas.
The solution to the "apparent" scourge of undocumented workers is to make it so that the people running companies the use that undocumented labor are the ones who go to jail, and in the event ICE finds undocumented labor the minimum fine for the company should be a significantly greater than one multiple of the amount the company would have had to pay a documented worker, in addition to ensuring the undocumented workers are fully compensated at the amount a documented worker would have been paid. Threatening to report them to immigration, underpaying, and wage theft are all clear evidence that the company knows they were employing people who were not allowed to work.
That's all that would be needed. Instead of ever more draconian penalties for the undocumented workers - many of whom have lived in the US for essentially their entire lives at this point, and simply don't have a choice - the penalties that should be being increased are the ones that apply to the employers. It's great because it will stop all that evil undocumented labor people claim to hate, because now these employers have to pay the undocumented labor if anything more than documented labor, because undocumented labor can report unsafe or illegal working conditions, come out fully compensated and in addition to those costs the employer has to pay even more in fines, while the managers and executives who knowingly employed said laborers go to jail.
That said as we've seen in Florida, plenty of businesses in America have set themselves up to be unable to operate without violating the law, so when Florida started passing its various "lets punish the workers, but not those that employ them" laws a whole bunch of businesses couldn't handle the idea of capitalism and complained about how they couldn't get any workers.