Precision on a thermometer for this isn't really important. If it's off by, say 2 degrees, even 5 degrees, you're still going to get the needed info about whether it is staying at a suitably cold temperature.
I would not turn the temp down more, instead make sure the door seals are in good shape and that everyone opening it, makes sure it closes good. Yes it uses more power, but also cycles the compressor more, wearing it faster. You should not have that big of an issue keeping the bottom at 0F, and having less than 10F difference at the top, unless your door seal is shot.
A fully packed deep freeze with a good seal, will have no problem keeping food at the top, completely frozen. It is far better to keep it packed full of food (or frozen water blocks, or even empty containers holding stagnant air if nothing else) so that each time the door is opened, there is far less exterior hot air exchange.
I highly prefer the ice blocks (or here, I grow tomatoes in summer and freeze blocks of those too) because they can keep food cold for several days during a power outage if you can resist opening the freezer much. It's also handy to have ice blocks, to just spontaneously grab and toss in a cooler for an outdoor event.