Even the smallest mini-splits are so ridiculously oversized for a small room in the back of the house that it would only cycle on/off, never modulate, and isn't worth either the money or the bother. The best "buy more equipment" solution would be smallest window shaker you can find.
The only exception where something the size of a mini-split makes sense as a single-room solution would be if the room was an unshaded west facing room with a lot of window area (say, a glass sliding door), in which case the peak loads might hit 3/4 ton mini-split territory.
It's worth recommissioning the duct system, by verifying that the ducts to that room (and all others) are well sealed with a temperature rated FSK tape (not duct tape) such as Nashua 324a (sold in box stores) or better yet, duct mastic on any seams or joints, and that the register boots caulked/foamed to the subfloor or wall/ceiling gypsum, and that the return is adequately sized for the supply, etc. In a reasonably balanced well functioning system a room that runs warm can be a symptom of duct issues, or duct design issue. If the ducts are in the attic above the attic insulation and the ducts aren't insulted to R8 that too could be performance issue. There may be flow balancing vanes in the duct system that have moved or were never really tweaked for maximizing flow to that room. Investigate that too. Beyond that, there may be a duct design tweak solution- replacing the ducts (both supply & return) with something slightly larger will improve flow, and as a DIY upsizing the duct might still cheaper than a window-shaker.
On hard-piping, seal the seams too (not shown in the pics) not just the joints.
Before you upsize any ducts, it's often possible (and better) to just attack it from the load end, and if that works, it's preferable to the noise & power-use of a window shaker in a kid's bedroom. If the attic insulation isn't up to snuff causing excessive heat gain through the ceiling it's worth bumping it up to at least R30-R38 with tightly fitted batts, no gaps or compressions. If there are south facing windows without awning/overhang or tree shading it's picking up quite a bit of solar gain during the day. West facing glass is even worse, since the sun angles are lower, making for even higher gain late in the day when it's already hot out, and difficult to manage with overhangs. Exterior pull down shades &/or heat rejecting retrofit window films can make a big dent in window cooling load factors anywhere roof overhangs or awning shading isn't possible.
Some combination of the above will work, but from a longest term, best comfort, and most satisfactory solution, attacking the load comes first, followed by improving/recommisioning the existing ducts, leaving buying more equipment a distant third.
If you don't already have one, a pistol-grip infra-red thermometer (~$50 at box-stores) is useful for figuring out some of this stuff. A ceiling that's 3-4F hotter than the room temp is dumping a heluva lot of heat into the rooms. Leaks in supply ducts can often be spotted as cooler spots in the surrounding areas. Low flow problems can show up as not-very cool temperature readings on the supply register grille compared to the grilles in other rooms, etc. Gaps in wall insulation (usually a 2nd order factor for cooling loads) can be spotted this way too.