In this case the answer is 'With'.
'the temperature falling so rapidly' is a noun phrase, so a preposition ('With') must go before it. ('For' is also a preposition but doesn't fit with the meaning).
'Since' or 'As' (=because) are conjunctions. This means they go before a clause (=subject + main verb in a tense), which is why we can only use them if we add the main verb 'was'.