Two filters must both be satisfied: the lesion has to be cancerous, and it has to push out both eyes in a child. That combination points away from a single solid orbital tumour and toward a blood cancer that seeds both orbits.
The entity that does this is the chloroma, also called granulocytic sarcoma, a tumour-like collection of immature myeloid white cells. Chloromas love the orbit and frequently appear on both sides, producing bilateral proptosis in a child. Since chloromas are built from myeloid blasts, their appearance is a hallmark of acute myeloid leukaemia.
Check the distractors against the two filters. Rhabdomyosarcoma is the most common primary orbital cancer of childhood but is typically one-sided, so it fails the bilateral criterion. ALL is the most frequent childhood leukaemia overall, but the chloroma-driven bilateral orbital picture is the signature of the myeloid (AML) variety. Lymphoblastic lymphoma rarely presents as bilateral childhood proptosis. Thus the bilateral, malignant, chloroma-based presentation maps to AML.
\[\boxed{\text{Acute myeloid leukaemia (AML)}}\]