Will an ice pack relieve my itchiness better than a cold compress?

No. The 2 are equivalent. Either one is fine. If it isn't working, Hydrocortisone cream on the itchy area or antihistamine pills (like benadryl) may help.
May need eye drops. Cold compresses are more tolerable than ice packs, and should be enough to provide relief of your itching. If this is ongoing you should consider using anti-histamine eye drops that your eye doctor can prescribe.