The best solution in this case is to stop means testing them, and also convert every plausible benefit from stuff to cash-to-buy-stuff-with (i.e. UBI). Because at that point it's the same as voting, all you have to prove is citizenship and that you haven't already received the benefit, there is no separate eligibility information required.
People have the intuition that not means testing things would be expensive, but when the benefit is in cash that comes out in the wash. If you receive $5000 more in cash than the value of the benefits you were previously eligible for, but then have to pay $5000 more in tax, it just cancels out and you're back to the original situation. Only now nobody has to prove eligibility status outside of basic citizenship, which also greatly reduces administrative costs and fraud.