I can find no record of 209th Antiaircraft Artillery (Searchlight) Battalion. 209th Coast Artillery Regiment (Antiaircraft) was later broken up to form Headquarters & Headquarters Battery, 209th Antiaircraft Artillery Group, 72d Antiaircraft Artillery (Gun) Battalion, 335th Antiaircraft Artillery (Searchlight) Battalion and 898th Antiaircraft Artillery (Automatic Weapons) Battalion. The above units served in North Africa and Italy and campaign credits include Tunisia, Naples-Foggia and Rome-Arno.
337th Engineers was a general service engineer regiment, not combat engineers. The regiment was disbanded in Italy in September 1944. Personnel were reassigned to Headquarters & Headquarters Company, 1338th Engineer (Combat) Group and used to form 169th and 182d Engineer (Combat) Battalions. 337th Engineers was likewise entitled to campaign credits for Tunisia, Naples-Foggia and Rome-Arno.
There was also a 337th Engineer (Combat) Battalion (formerly 337th Antiaircraft Artillery [Searchlight] Battalion which served in Italy and was transferred to the Philippines shortly after the war ended in Europe. This unit received credit for the North Apennines and Po Valley Campaigns (337th Antiaircraft Artillery [Searchlight] Battalion was credited with participation in the Rome-Arno Campaign).
"Battle of the Ro-Marno" appears to be a corruption of Rome-Arno Campaign.