Why not institute a guest worker program that requires a state permit (and maybe a federal one too while we are at it) ?
Make the state permit for CA $100 monthly for any month in which a worker is used, paid for by the company.
That’s 1200 per year per worker. If there are 4 million guest workers, this would approximately cover the $5B shortfall. If not, scale the fees up.
Make the fines for not registering a factor of 10 or 20x higher than the permit and generate additional revenue. Use this additional revenue to pay for the enforcement, e.g. through a bounty system. I am sure that some of the workers would report the employees to collect the bounty, thus forcing the employers to comply with the permit process.