This is a translation which can often cause problems. So let's work backwards and go by the needle size and see if that helps.. 4mm is a US 6 and 2.25 mm is a US 1.
They are using the US 6 on the fun fur and then switch to the US 1. A US 1 is usually used with thin yarns so probably fingering weight would work. That's what you are using now with the #1 designation.
This is a fitted dress so it's supposed to be rather tight, but if you find it too tight you can go up a needle size. That's usually the best way to do it rather than changing yarns.
Thanks for the link btw. That's a lot of Barbie clothes!