While I don't doubt the P320 influenced the P365, the P365 wasn't submitted for the recent service pistol competition and I'm not aware of a police department that issues the P365 as the primary duty weapon (maybe for plain clothes officer or detectives). This isn't me saying there is no difference between the LCP and the P365, I believe there is.
The SIG SAUER P365 gives Indiana State Police Troopers primary carry capabilities in a sub-compact configuration,” says Matt Bilkey, Lt. Colonel, Assistant Chief of Staff Operations, Indiana State Police. “Its high level of accuracy, small size, and high round capacity make it a perfect firearm for our officers, both on and off duty, and is a great counterpart to our primary duty firearm the SIG SAUER P227.
https://www.policemag.com/510876/in...-sig-sauer-p365-as-their-back-up-duty-firearm
As for our detectives, the P365 is the perfect pistol because of the small size and high-capacity which makes it easy to conceal without sacrificing rounds.
The Sumter Police Department is our first agency to simultaneously adopt both the P320 and the P365 pistols as their official duty firearms and we could not be more excited about it,"
https://www.policemag.com/363154/south-carolina-pd-transitions-to-sig-p320-and-p365-pistols