There are two categories pertinent to the understanding of the subject in this type of sentence: simple vs complete; and grammatical vs real.
Your sentence happens to be a copulative sentence, which has the following three parts, ordered usually in that fashion: SUBJECT + copula (be/get/seem/etc.) + PREDICATIVE EXPRESSION. So, yes, the subject is the entire clause whether or not to bomb Syria.
It may not look like a typical clause, but it's a clause, a so-called infinitive clause, and those really are clauses, because they may contain such clause elements as a subject, complement (by that, I mean indispensable modifiers), and adjuncts (dispensable modifiers). (In your case, Syria serves as the object complement.) Infinitive is a non-finite verb and the only clause in which it can be the main verb is a non-finite clause. Non-finite clauses are never independent clauses. That yours is dependent can also be surmised from the presence of the word whether, which is a subordinator. But let's move on.
A complete subject tells what or whom the sentence is about. A simple subjects is the main part of a complete subject and thus is free of any modifiers. Both point to what or whom the sentence is about. Also, both can comprise more than one word. Now, you may think that in your sentence the simple subject is the infinitive to bomb because we all known that infinitives can act as nouns. You'd be wrong. A sentence To bomb a country means to have a war on one's own back. does have to bomb as a simple subject, as a part of the complete subject to bomb a country. But your sentence is not about bombing (a country, or Syria) per se. Rather, it's about the question (whether or not to bomb (Syria)). So, the entire clause whether or not to bomb Syria is the simple subject. On a grammatical level. The grammatical subject. You can look at it this way: all the words of that clause are nothing but an explanation of this concept-identifer: question. Which is a noun. So, in an appropriate context, the following sentence would have the same meaning: The question is the pertinent question. Additionally, the following sentence would be said to have a clause as a complete subject: Whether or not to bomb Syria, earlier unthinkable, is the pertinent question.
Let us now move onto the grammatical/real split, because in conducting a syntactic analysis, lay analysts ordinarily make a mistake of letting this split influence their thinking, beffudle their mind. OK, having moved to that one level up in the analysis, we can observe that in the actual world, to bomb something (or "bombing" something — gerunds are similar to infinitives) cannot be agentless. If a country is to be bombed (nota bene, passive infinitive), somebody is to bomb (active infinitive) it. Who? The real subject, that's who. Now, grammar allows us to passivize an active expression while keeping the same meaning, and vice versa, so that will make it easier to find the real subject. Let's passivize it:
whether or not to bomb Syria -> whether or not for Syria to be bombed
(We can afford to insert a for, because for doesn't carry much meaning as yet another subordinator.)
But, luckily, passives always have the "by" adjunct, even when it is merely implied and omitted. So, the complete passive version is, actually, this:
whether or not for Syria to be bombed by us
But, this is still a non-finite clause, the infinitive (even when passive) has no tense and no grammatical category of person, so let's forcibly make the clause finite. It's OK, we're standing more in the semantic demesne than in the syntactic one. We'll go with the dummy it as a subject and the verb to be, because both are skimpy on meaning, compared to other words from their class.
whether or not it is for Syria to be bombed by us
Let us now depassivize that extended clause:
whether or not it is for us to bomb Syria
This means nothing but this:
whether or not we should bomb Syria
If we omit the subordinator to make the clause independent:
we should bomb Syria
And there's your real subject: we. Here's more on the grammatical/real split.