I wouldn't use a systemic insecticide on edible plants of any kind.

I know what you are saying rhizo, I feel leary of it, too, but the reality is that the commercial (non-organic) growers use it widely on edible plants. So unless you only eat organically grown vegetables/fruits/grains, you are exposed to it. It is considered non-toxic to mammals but who knows what future research may show regarding long term exposure.

I suppose I need to just hope for the best since the contact insecticides aren't effective against the leaf miners.

You can simply squish the leaf miners in the leaves when you first notice them. Or remove those leaves -- or the damaged parts of the leaves (being sure to bag and discard).

If I squash the leaf miners or remove the leaves, won't that do as much damage as the leaf miners themselves?

If you squish the leafminers, you damage a bit of the leaf, but save the remainder of the leaf from being mined. I imagine it's best to be careful not to damage the major veins.

If you remove (then bag and get rid of) the damaged leaves before the leafminers have pupated, you eliminate the next generation of miners.

In a thread earlier this year, planatus said it was safe to remove up to 30% of a tomato plant's leaves (in case of damage of one sort or another). [I can't believe I actually located that thread!]