Do you have a good supply? I think if you express the first 15-20ml this is supposed to help a bit, since the foremilk is a lot higher in sugar than the more fatty milk.

You can try drinking other sorts of milk if you want to go off dairy. Soy would be the most common, but a lot of babies with dairy sensitivities have them to soy too - if its just the lactose though, then it may be fine.

you can also buy rice or almond milk at wollies/safeway and coles - not sure about smaller stores, although a well stocked health food may have something.

Cheese wise - speaking as an ex-vegan, I found no nice nondairy cheese (at least for eating on its own) at all while I was off dairy