"lacto-free milk", but I'm not quite sure enough exactly what she's allergic to to know whether that would be ok.
Yeah, I'd go with "don't touch lacto free milk with a bargepole unless you know for sure"

You could probably make an educated guess on what is most likely the reason based on e.g. whether or not they ate any other dairy products, or if they seemed like they had a cold all the time, or if they had asthma, or which part of the world their ancestors came from (we are biased by living in Northern Europe to think that adult lactose tolerance is the norm, but worldwide it isn't).

But best not try the guess on them ;-).

Will it still be palatable?
Ah, but would it be palatable in the first place?! ;-). I don't put fake milk in my coffee (unless I'm making a latte, where I start from not-cold milk), because it curdles and tastes less nice than black coffee unless you handle it just right.

Can you recommend a non-dairy milk-esque product which comes in smaller boxes/cartons/quantities?
If you go to a big supermarket or a healthfood store, you can buy small cartons of not-milk to be drunk with a straw, like you would have got in primary school if Maggie hadn't got rid of them. Many of them are long life. You can also get powdered soya milk which you make up, though I thought the concept was too horrific to try.

And I don't want to ask her about this in case it sounds like I'm finding it a big problem. Which, of course, it isn't - soya milk is cheap - it's just a bit wasteful.
Could you not just drop it into conversation casually, along the lines of "Oh, my friend Becky says that $otherMilk is tastier, what do you think?". Or even "My friend Becky thinks coconut is indigestible, isn't that strange and unusual?"

That way you can find out their favourite brand ;-).