More on the Patria -
Historical Background
These were Jews who were allowed to leave German controlled territory a year into the war. Keep in mind, it was the British (Churchill) who was trying to cut off the escape of the Jews from the Nazis, not the Nazis themselves. The Germans would turn to extermination only when the British efforts proved successful. The ship itself held refugees from several other ships that had been detained by the British. The British packed the ship to about 2.5 times its capacity, contributing to the disaster (without making provision for additional lifeboats).
The explosion
The explosion was not carried out by the Nazis, or even the Muslims - it was carried out by Jewish Palestinians under the direction of the Haganah. They apparently misjudged the size and placement of the explosives. The purpose wasn’t to kill the central European Jews, but rather to disable the ship so that the refugees couldn’t be transported to British camps in Mauritius.
The action must be deemed a success, as well over 1000 survivors were allowed to land (although they were initially held at a concentration camp in Palestine). 260 were killed, and over 170 were injured by the blast or from being trapped below the waterline. On the other hand, a similar number of refugees on another captured ship were transported to the camps in Mauritius.
For context, as of "today" the Nazis are still considering transporting Jews to French Madagascar.
As you say, they turned to extermination after other options for a "final solution" seemed impossible.
From the British perspective, allowing more Jews into Palestine would put their friendly relations with Arabs at risk, and make it more difficult to defend the Middle East against future Axis invasions -- in North Africa, for example.
British fears of Muslim reactions might seem overblown, until you consider that many British Commonwealth units came from Muslim countries.
So it wasn't just a matter of Arabs opposing the British military.
It was also a question of the loyalty of some British units.