Banning double jeopardy laws will not affect the integrity or effectiveness of police work. If anything, it will be a positive movement. The fact that the police have "one shot" in which to provide a successful investigation will not wane on the off chance that they will now have a second chance in the future for conviction.
Unfortunately, some police officers and departments aren't known for their integrity. Still, double jeopardy rules promote integrity rather than putting it on the back burner. Getting rid of these laws would give police officers no reason to exhibit integrity during their investigations. Therefore, such laws need to remain in place.