Can you still "Push" (DC 25 Handle Animal), your dominated Animal Companion while its under the control of an enemy caster with Dominate Animal?
I would rule no, though the rules do seem ambiguous. My reasoning comes from this line in Dominate Animal: "Because you are directing the animal with your own intelligence, it may be able to undertake actions normally beyond its own comprehension." The implication from that line is that, while the outside is all animal, the mind isn't, since an element of the dominating caster's own mind is present.
The closest RAW ruling I can think of comes from some information on multiple mind control spells on the same target. Essentially you would have to make an opposed Charisma check (in addition to the Handle Animal check to assert control) with the dominating caster.