The circuits all over the place on this one but in short, no, police are not obliged to apprehend a suspect at the earliest opportunity.
It is within the discretion of the police to decide whether delaying
the arrest of the suspect will help ensnare co-conspirators, as
exemplified by this case, will give the police greater understanding
of the nature of the criminal enterprise, or merely will allow the
suspect enough "rope to hang himself."
U.S. V. Garcia 79 F.3d 74 (7th Cir. 1996)
See also Hoffa v. United States 385 U.S. 293 (1966)
A suspect has no constitutional right to be arrested when the police
have probable cause.
The police are not required to guess, at their peril, the precise
moment at which they have probable cause to arrest a suspect, risking
a violation of the Fourth Amendment if they act too soon, and a
violation of the Sixth Amendment if they wait too long. Law
enforcement officers are under no constitutional duty to call a halt
to a criminal investigation the moment they have the minimum evidence
to establish probable cause, a quantum of evidence which may fall far
short of the amount necessary to support a criminal conviction.