Dairy-free bechamel?
Butter:  Butter is an emulsion of fat and water (more or less)—it is typically around 80-82% fat (European butters tend to have a little more fat, American butters tend to have a bit less)—and this fat is solid at room temperature.  This suggests that a good substitute for butter would be another solid-at-room temperature fat (such as coconut oil, lard, bacon grease, etc), plus a little bit of water.
Personally, I have had luck with coconut oil and a splash of coconut milk when cooking for vegan friends, but this leaves the sauce tasting a little coconutty.  Lard or bacon grease will similarly flavour the sauce, but this may not be a negative—macaroni and cheese with bacon, for example, is a great combination.  Olive, canola, and peanut oils have also worked for me in the past, but result in a slightly different texture at the end of the day (which you likely wouldn't even notice in a casserole like moussaka or lasagne).
That being said, butter has very little lactose in it.  A lot of people who are lactose intolerant do not have difficulty with butter.  It might be worth experimenting with your child to determine whether or not butter is a problem (e.g. go through the process of an elimination diet, and add butter back into the diet in a controlled manner to see if is tolerated).

Milk:  As noted above, I have had good luck with coconut milk.  Because the phrase &quot;coconut milk&quot; can be ambiguous, I specifically mean something like Thai Kitchen's unsweetened coconut milk.  It should be noted that this has a much higher fat content than dairy milk, so you may want to water it down a bit.  Nut milks, soy milk, or oat milk should also work well—you are really just looking to get something that is mostly water with a bit of fat in it.
You could also consider lactose-free milk.  There are a number of commercially available brands of lactose-free dairy milk, which are made by treating milk with with the lactase enzyme (this is an enzyme which breaks down lactose, and is produced naturally by most people—lactose intolerance is generally the result of a person's inability to produce lactase).
Another alternative would be to consider the milk of another animal—my understanding is that there is some anecdotal evidence that goat's milk is often better tolerated by people with mild lactose intolerance.  Goat's milk has about as much lactose as cow's milk, but is apparently otherwise easier to digest.  Again, experimentation with an elimination diet might prove fruitful.  On the other hand, goat's milk has a rather distinctive flavor which many people find off-putting, so maybe not.

Cheese:  I have never found a non-dairy cheese which I thought was any good.  There do exist vegan cheeses, some of which even seem to melt like dairy cheese, but I (personally) find them all to be quite bland and unappealing.  They compare favorably to Kraft singles, but that seems to be damning with faint praise.  That being said, taste is personal, and your milage may vary.  Vegan cheeses are meant to replace dairy cheese in a one-to-one manner, so if you find a vegan cheese you like, it should
On the other hand, cheese often has far less lactose than milk, and the longer a cheese has fermented and aged, the less lactose it is likely to contain.  I am not sure what to do about the ricotta in a lasagne (assuming that you use ricotta in your lasagne), but the parmesan should be well-tolerated as it typically contains almost no lactose.  For other cheeses, as a rough rule of thumb, the older and harder a cheese is, the more likely it is that it will be well-tolerated by a person with lactose intolerance.
Again, an elimination diet might be in order.