Gluten/ lactose free alternative

Thank you so much for the responses. That's really interesting about butter- I wasn't aware there was a very low amount of lactose in it. I've seen my friend (who I'm baking for) ask for no butter on food when eating out so assumed this was related to being lactose free, but may just be a preference.

I asked at very short notice before going to shop so ended up buying a block of Stork margarine. It seems to have worked really well judging from the look and smell alone.

Thanks for the comment about other recipes too- searching for paleo cakes is a really good idea as my searches for gf and lf kept bringing up cakes which didn't meet both criteria.

This is an amazing cake.
https://leitesculinaria.com/108297/recipes-paleo-chocolate-cake.html

Agree with KLS advice about butter or ghee.
Another way I go about making allergy-sensitive versions is to search for an existing recipe for a similar item. That way, other people have already road-tested the replacement ingredients and proportions.
If you go that way it won't be Emiko's exact recipe, but it will be similar and good.

Butter is very low in lactose, 0.1 gram per 1 cup. MOST people who are lactose intolerant can tolerate butter because of this small amount of lactose.
That being said, if your friend has a high sensitivity, that could still be a problem.
Some of the newer plant based butters could work, but I think the best could be ghee, which is clarified butter, the milk solids are removed.
Let us know if you make it.