Interesting hypothetical. I'll do what I can to run with it.
First, as others have said, mixing police and army = Trouble
I don't know what the population, or pop density of this island is, but working with what we have, I would detail off about 500 of the force into permanent police duty. One of the OP's criteria was that this island has no gun culture, and has just been granted independence from a larger neighboring nation. I'm going to expand that to assume that they have been under an authoritarian (more or less) regime, that there is commensurate crime, and maybe some sort of internal friction. I'm thinking "Sri Lanka", only smaller. So there needs to be a police force until crime is brought under control, and we start embracing personal freedom as a culture. Or not, since I'm apparently the dictator of this little island.
So my 500 guys will be detailed to police precincts that I would be establishing. Probably I'd end up with 250 or so in the urban area, 50 in the rural areas, and 200 doing coast guard duties (it is an island, after all).
My 250 urban cops get broken up into five platoon-ish sized units, with four of these being responsible for sections of the city, patrolling, responding to calls, et cetera. My fifth unit is a mixture of SWAT, Aerial, and Vice. This unit may expand or contract as necessary. The beat cops get nightsticks and Kel-tec P11s. At 200 x $333 = $66,600 to arm. The other fifty would be armed as needed; aerial might not be armed at all. Vice would probably use seized weapons, to blend in with criminal armaments. SWAT would have appropriately modern armaments, including Remington 700s for sniping, (about $700 + 200 for glass, X 5 rifles total, = $4500), Remington 870s for breaching ($360 for 18-inch synthetic model, X 5 = $1800), and Kel-tec Su-16Ds for the shooters ($874 x 10 = $8740). Pistols for SWAT are optional (encouraged), paid for with their own salary.
Rural cops get Kel-tec Su-16As ($665 X 50 = $33250). Like SWAT, if they want a pistol, they can buy one.
Coasties are a bit of a problem. I'd probably have a lot on cutter-size vessels, and a lot on river/harbor patrol in motorboats, and maybe another aerial unit. For convenience' sake, I'll say I have to arm every other coastie, since the cutters would probably just have an armed boarding party among the crew. I'd arm them with a mixture of Kel-tec Su-16As and Remington 870s, at about a 9-1 ratio. (90 X $665 = $59850, 10 X $360 = $1800)
The other 500 is split into two groups. 250 or so get detailed to administrative and maintenance duties. Payroll, processing, quartermasters... motorpool duties for cop cars, boats, aerial. No armaments. They are, however, free to carry personal weapons at work.
The final 250 is the "army". 200 get silly uniforms and partisans (short pikes). Yes, they are my papal zouaves. They march around and look pretty. They stand impressively motionless outside of government offices, put on parades, man the gates at my enormous palace, serve as honor guards, and so on.
The remaining 50 are highly trained, very well armed, special forces teams. They do anything that needs doing, as long as it's violent and difficult. They would get whatever they need. Their weapons budget is $250,000, and they get access to automatic rifles, squad automatic weapons, RPGs, or whatever. I wouldn't mandate their armament in any way: They have a toolbox of weapons, as well as use of any government vehicles, and can modify the latter at need. These guys are my insurance. I pay them directly, and very well. Their job is to keep me in power, head off any home-grown coup, and deal with messy problems the cops can't handle. These guys are all volunteer, we accept foreign recruits (but keep the proportion low), and their psych profiles are vetted, backgrounds checked. If things go well, nobody even knows these guys exist.
So, my weapons budget comes to:
$176,540 for police and coast guard arms
$250,000 for my household of ruthless, fanatically loyal killers
The other $573,460 I use to buy LMGs, RPGs, explosives, and some small arms. These go into a armory underneath my palace. If we're invaded, I break these out and hand out small arms to my zouaves, and heavy weapons to the cops, so they can help "repel boarders". Despite being a despot, I encourage civilian firearm ownership, so that my people can also help defend their little island.
Personally, I think I could stay in power for a whole 3 or 4 days. Maybe even a week.
Edit: Forgot ammo. I guess I'd drop half of that remaining $500k or so into ammo. Maybe I should just make everyone pay for their own, though...