As soon ... as find
No, it's not the same sense as as soon as possible, it's would (just) as soon. https://dictionary.cambridge.org/dictionary/english/would-just-as-soon
They are as ready to take action A as they are to take action B.
This is an idiomatic usage of as soon...as.... It has the same meaning as rather...than... or sooner...than... (CaGEL p1133). The approximate meaning of the sentence is:
the local constabulary prefers dropping them in a communal grave to finding a name for a gravestone