Did Ellen White say children shouldn’t touch dogs and cats?

I agree with you that Leviticus 11 is directed toward what one eats. This chapter says that people who touch the dead body of an unclean animal are made unclean for a time. I know of no such warning about the animals while they are alive. The camel, for instance, is named in verse 4 as unclean, but God’s people were not forbidden to own camels, ride them, etc., which must surely have included touching them. The same principle applies to horses and donkeys. You can’t care for such animals or use them productively without touching them.

I know of no statement from Mrs. White that forbids children from having pets such as you describe. Mrs. White herself owned a dog while she was in Australia. It was a watchdog whom she named Tiglath-Pilesar, probably because he could be fierce toward those he might consider possible enemies. But Mrs. White seems to have felt affection for him, even to giving him that biblical name. I think it must have brought a smile to people! Another indication that the dog may have been special to her is that her friends in Australia devoted a whole page to him in a picture book they gave her when she left for America. Here is what Arthur White wrote about the matter in Ellen G. White: The Early Elmshaven Years, 1900–1905, 5:19:

The brown-toned photographs help to tell the story of the work in Australia. There is the electro-hydropathic institute in Adelaide. There are pictures of neat little churches Ellen White had visited and in which she had made investments to help the companies of believers who needed meetinghouses. There are portraits of friends, and scenes from her Sunnyside home. One page was reserved for pictures of their watchdog, Tiglath-Pileser, at Sunnyside. It will be remembered that parts of Australia had been settled by convicts, and as some of their descendants seemed to inherit the proclivities of their forebears, a good watchdog served a very useful purpose at Sunnyside.

So I find no basis in the Bible or in Mrs. White’s writings for forbidding anyone to touch a cat or a dog because it is an unclean animal.