I will offer a different perspective, and argue that under the right circumstances a policing military may function with greater professionalism than a civilian police force. In fact, I will argue this is already demonstrable in America...
It's worth prefacing that the benefits of this are also drawbacks. The military has more authority and less regulation on its behaviour than civilian institutions, but given the right objectives and ethics this may not matter much.
Stephen Mader was a US Marine Corp veteran who had served in Afghanistan. He then quit the military and joined the police. One day he was called to a domestic incident, and had been told a man (Ronald D Williams) was outside waving a gun about. Along the way he got an update that the man's girlfriend had called. She had tried to explain that her boyfriend was depressed and suicidal, and had created a scene to invoke suicide by cop.
Mader arrived at the scene and in this moment relied upon his experience from Afghanistan. In Afghanistan he learned to assess whether the people passing him in the street wanted to kill him or not. Soldiers in Afghanistan were instructed to use caution before resorting to violence, otherwise mistakes could happen which would push more Afghanis into the arms of their enemies.
He saw Williams waving a gun about for show, and didn't see anything in Williams' behaviour to suggest violent intent. Mader tells Williams to put the gun down, so they can talk.
Mader's colleagues then arrive to provide backup, and they make a shallow assessment. Here is a civilian waving a gun about, threatening police officers. They tell Williams to drop the gun. He doesn't and they shoot him. He dies from his injuries. Mader is then fired for 'endangering' his colleagues by refusing to shoot Williams.
The US military in Afghanistan evolved much like the British army did in Northern Ireland, learning that brutality often backfired and that they needed to operate carefully to avoid inflaming resistance against them. A military that has evolved in response to such circumstance would place a high value on caution and reading people. Consequently they may operate with greater professionalism than a civilian police force which does not need to worry about a popular backlash from being trigger happy.
The situation you describe sounds a bit like Northern Ireland's Troubles; a war which comes and goes, the military always watchful. Although maybe I'm just reading that into it, since you say most of their duties will be breaking up drunk fights.
Incidentally, there's an informative podcast where Sam Harris interviews Jocko Willink, a former navy SEAL who was deployed to Iraq. It contains an interesting discussion on the idea that training for violence, with martial arts especially, helps to reduce the chances of a needlessly deadly escalation. Inferring that soldiers given proper training may lead to less deadly outcomes in civilian confrontations precisely because they are more confident of their ability to inflict deadly violence, and thus are less willing to use it... admittedly this could be said for military or police.
Another benefit of combining the police and military might be far more extensive intelligence gathering operations. The military can just create watchtowers and road blocks and various other check points wherever they please because they operate above usual civilian oversight. They also will be able to use spies in a way the police usually wouldn't, creating extensive networks to attempt to control subversive elements. Like Britain's FRU, or Pakistan's ISI. But this can get out of hand for obvious reasons.
In conclusion, I'm going against the popular feeling. Policing with the military is a great idea* for the aforementioned reasons, and offers many unique benefits given the right context. After all, there's already paramilitary police forces operating under military leadership in many countries without problems, such as France's National Gendarmerie, or Italy's Carabinieri. Gendarmeries are actually remarkably common around the world.
( * This is not an endorsement of using the military to police civilians. )