Is it permissible to recite Ayatul-Kursi for protection without the Basmala and Isti'adha?

If you recite Ayatul-Kursi for ruqyah, or any other part of the Quran, you should not omit seeking refuge with Allah from Satan, due to His saying: ﴿So when you recite the Quran, seek refuge with Allah from Satan, the expelled.﴾

As for the Basmalah, it is prescribed at the beginning of Surahs. So, whoever recites from the middle of a Surah should seek refuge and not say the Basmalah. Whoever recites from the beginning of a Surah should say the Basmalah after seeking refuge, except at the beginning of Surah At-Tawbah, where the Basmalah is not prescribed.

If you recite without seeking refuge or saying the Basmalah, there is no sin upon you, because seeking refuge is recommended, not obligatory. However, it is advisable to do so and to say the Basmalah if you recite from the beginning of a Surah.