Q: Brahman is beyond cause and effect. But a seeker’s effort is bound by the principle of cause and effect. Therefore, it would seem that a seeker cannot  reach a state of Self-realization, howsoever much his efforts are sincere.

Obviously, this cannot be quite right. After some though, I arrived at the following possible explanation:

The Brihadaranyaka Upanishad, for example, has the mahAvakya: aham brahamAsmi: I am Brahaman.
Brahman+ avidyA = jIva.
If the jIva makes an effort ( which is subject to the cause and effect principle) to remove avidyA, then it can be removed. Once avidyA has been removed, then the ever present, self-luminous Brahman is re-gained. So the method is to remove avidyA. 

Is my reasoning correct?

A: It is not quite correct.

Your main problem is in confusing absolute and empirical reality.  Brahman is the absolute reality; time, space, causality have no meaning, since absolute reality is non-dual. The seeker IS Brahman in absolute reality. At the empirical level, of course, the seeker lives in a world of apparent duality and is subject to cause and effect.

Secondly, what you say implies that you are unclear about ‘Self-realization’. Rather than use that term, or ‘enlightenment’, it is better to refer to ‘Self-knowledge’. Then it is clear that ‘Self-knowledge’ removes ‘Self-ignorance’. If you make the effort, you can remove Self-ignorance. When this happens, you realize that you are Brahman, and always have been, since there is only Brahman. The effort relates to the jIva in the world. The Self-knowledge relates to the mind of the jIva in the world. All these are mithyA, since there is really only Brahman.

So yes, if you make efforts (subject to causality), you can remove avidyA. Then you do not ‘gain’ or ‘regain’ Brahman; you simply realize that you ‘are’ Brahman.