There is no such nation as Amalek today. The descendants of Amalek are dispersed among all nations, including the Jews themselves.
According to one halachic opinion. The Religious Nationalist camp of rebbes would disagree.
So you want to state that according to other halachic opinions "Palestinians" considered to be Amalek? Please quote them. I think you'll find it hard because such halachic decrees do not exist.
But deliberate killing of so-called "Palestinians", no matter what age, JUST BECAUSE THEY BELONG TO THAT NATION is absolutely forbidden by Torah Law.
First of all there is no Palestinian "nation", but even if there was you would be wrong. This statement almost opposes Zionism himself. What did G-d command the Israelites to do to the Muslim Nazis of ancient times (Canaanites, Amalekites, Midianites, etc.), the Nazi beasts whose satanic moon-worship religion directly spawned Islam?
1. I know there is no Palestinian nation. This term is forged by Arab propaganda to oppose Zionism and the right of the Jews to the Land of Israel. That is why I write it using "" just to avoid long description like "Arabs currently living in the Land of Israel".
(I have to admit it is a very powerful term to appeal to the Christians because for many Christians it is a synonim for "Holy Land", "the land where Jesus was born", etc. so when Arabs are fighting for "Palestine" against the Jews they imply that it is in Christians' interests too).
2. G-d indeed commanded to exterminate (or let peacefully go if they choose to) ancient Cannanites and other sinners but it has nothing to do with our times and with Islam. You do not have any halachic basis to link between two these things. Judaism is law, not poetry and parables.
The fact that G-d commanded Abraham to sacrifice Isaac does not mean that every Jew or righteous gentile should do that. It was one-time commandment to the specific man. The fact that Pinchas killed Zimri and his Moavite girlfriend while they were committing adultery does not mean that every Jew or righteous gentile should do that.
Each person and generation have their own challenges.