I want to know whether someone would have been likely to use the expression just before the first world war.
Can anyone help?
From the following Ngram you can see that there were no recorded uses before 1900 and then no uses of your best bet until about 1920. Unfortunately recorded usage in books is always slightly behind spoken language and Google Ngrams only records books. Nonetheless, I'd opt for no, I doubt someone before the war would have used that expression.
safest or most reliable course of action :
surest means to a desired end :
most advantageous approach :
most satisfactory choice : the pilot's best bet was to make an emergency landing
the best bet for stabilizing the national economy
First Known Use of best bet 1906
At the same time as BladorthinTheGrey was looking up Ngrams I was doing the same thing, only I used a tighter set of search criteria. This shows that 'best bet' has a few mentions in American English publications from about 1908 onwards but a similar search revealed no usage in British English until 1931. It then disappears from British English publications around 1935 and only reappears about 1950 onwards.
This suggests that 'best bet' was a newish term in the US in Edwardian times and was very uncommon or completely unknown in the UK until after WW2. If your characters are young Americans you might get away with it but if they are British it's completely inappropriate.
Doing an Ngram search for 'best chance', however, shows this in use from the early 19th century in both the UK and the US. This might be a better choice but, probably, in the form of "the best chance you have" rather than "your best chance" as the latter returned no Ngram results at all.