'phh' - You did locate the cross-pin. Drive it Left-Side of rifle to Right.
A scrap metal piece that will fit in the sight-base, (where the 'movable-base' traverses), will give you a surface to pound on. This protects the delicate base pieces from being bent or broken. Some heat on the 'collar' can help loosen it from the barrel, before you dive it forward.
You can do this with the barrel still on the action. (The second pin will just go forward with the sight-collar. The pin may be on the right or left. In my opinion it is not necessary in a rebuild. A little Epoxy will work just as well).
1903 barrels are easier to remove than M-1917 barrels.
Good Luck on your noble project!