If they are worried about inspectors, the don't need to see dead mice.... mouse droppings are just as incriminating for evidence. Tin cat style traps along all edges would be fairly effective, as baited traps will not be very effective in a structure with available food.
But any grocery store with a mouse problem should call an experienced PCO, and targeting with best poison would be the likely answer. I'm not a poison fan, but in structures such as this, restaurants, or any public business or government structures it's usually the best method to knock them down fast.
The follow-up exclusion work (mouse-proofing) would be the next step, and most pest control companies do NOT offer the service. Their business model is based on repeat contracts requiring monthly or quarterly visits. They usually just treat the symptom instead of actually solving the problem.
You may be able to have the pest company knock the numbers down, then let the company know you could do the exclusion work to eliminate (or minimize) the chance for reinfestation.
As mentioned above.... you could do heavy trapping programs from Friday evenings through early Monday mornings, maybe 15 to 20 of the rectangular repeating traps (tin cat style). Hit them hard during the time period that inspectors are home watching TV for the weekend or playing golf with their buddies from the public health department.
