A room built over an attached garage is the single most common problem room in American housing, and a ductless head is the usual remedy.
Why it behaves badly
The floor is over unconditioned space. The garage below is not heated or cooled, and in summer it is hot and in winter it is near outdoor temperature. That floor is an exterior surface, and it is frequently poorly insulated because the insulation is hard to install and easy to install badly.
The ceiling is under a roof, often with knee walls and sloped sections that leave small triangular attic spaces on each side. Those spaces are unconditioned, poorly air-sealed, and reach extreme temperatures.
Three or more walls are exterior, sometimes with gable-end glass.
So a room that looks like an interior bedroom has five exterior surfaces where a normal bedroom has one or two.
It is at the end of the duct run. Whatever central system serves it, this room is the furthest branch, and it gets whatever air is left.
Why more capacity from the central system does not fix it
Adding capacity at the equipment does not change how much air reaches the end of the longest branch. The room stays short.
Nor does a bigger central system change the fact that this room's load per square foot is several times the rest of the house's — a single thermostat elsewhere cannot serve both.
What a ductless head does well here
It puts capacity in the room, independent of the duct system, controlled at the room.
It is the correct mechanical answer, and it is why so many of these rooms end up with one.
Sizing it
Do not use the standard 20 BTU per square foot. This room's load per square foot is much higher, and a head sized by area alone will be undersized.
A room load calculation is worth doing properly here more than anywhere else in the house, and it needs the floor assembly, the knee wall areas, the sloped ceiling assembly and the glass by orientation.
Expect a figure well above what the floor area suggests — and note that once the envelope work below is done, the figure drops sharply, which is an argument for doing the insulation first.
The insulation work that should come first
Air-seal and insulate the floor over the garage, ideally with the cavity fully filled and in contact with the subfloor. A batt sagging in a joist bay with an air gap above it does very little.
Air-seal the knee walls, and treat their back faces as exterior walls — they need insulation and an air barrier, and the triangular spaces behind them need to be separated from the room.
Seal the floor cavity at the knee wall, which is a direct path from the attic space into the floor structure and one of the largest leaks in the assembly.
Insulate the sloped ceiling with a ventilation path maintained above it.
Done properly, this changes the room from the worst in the house to an ordinary one, and it reduces the head size needed.
The order
Envelope first, then size the head, then install.
Doing it the other way round produces an oversized head in a room that no longer needs it, with the humidity and cycling behaviour that follows — and a bonus room is exactly the kind of space where that is noticeable, because it is often a bedroom or an office where people spend hours.
The garage below is part of the problem
An attached garage is not conditioned and is frequently not air-sealed from the house, and in summer it runs far above outdoor temperature with a car parked in it.
Sealing and insulating the garage ceiling is the work described above. Insulating the garage itself is a separate and optional step that lowers the temperature the floor above is exposed to, and it is worth considering where the garage is also used as a workshop.