I found one 2018 study finding that policemen who were former military veterans had fired their weapons more often than non-veteran policemen. The study does have a somewhat narrow geographical scope...
Researchers at the University of Texas School of Public Health in Dallas examined the U.S. Armed Forces records of officers in the local police department and published their report in the Journal of Public Health on Oct. 3. They billed the study as the first of its kind and found that regardless of their deployment history, cops who were military veterans were more prone to shooting incidents. Officers who were deployed were 2.9 times more likely to have fired their weapons while on duty. [...]
“Results from this study suggest that veteran status, regardless of deployment history, is associated with increased odds of shootings,” the professors wrote. “There is critical need for additional studies on the relationship between veteran status, deployment history and combat experience, and officer-involved shootings.” [...]
In total, 516 cops—with and without U.S. Armed Forces experience—were examined. Close to one-third of officers involved in a shooting had a military background, researchers found. Military veterans made up 16 percent of officers who had no shooting incidents.Though 6 percent of the general population has served in the military, 19 percent of police officers are veterans, according to an analysis of U.S. Census data performed by Gregory B. Lewis and Rahul Pathak of Georgia State University for The Marshall Project. Policing is the third most common occupation for veterans, behind truck driving and management.
(Eventually) found a link the actual study https://pubmed.ncbi.nlm.nih.gov/30281075/
Also, conducting such research in the US is fairly difficult. From an earlier article from the same source:
Official data on the impact of veteran-cops is scarce. Nearly all of the 33 police departments contacted by The Marshall Project declined to provide a list of officers who had served in the military, citing laws protecting personnel records, or saying the information was not stored in any central place. The Justice Department office that dispenses grants to hire cops and study policing said it has no interest in funding research into how military experience might influence police behavior.
Since someone suggested in a comment that the ex-military are more likely to be SWAT which might explain the increased involvement in shootings, that turns out not to be the case in this study. Patrol and SWAT assignments were not (statistically) distinguishable in this regard (shootings) although administrative assignments were (much less likely to be involved in shootings).