The mini-split units are quite expensive compared to other solutions, but they are nice.
To consider:
The "booster fan" in a duct will help move cool air to the hot room when the main house AC is running. When the main house AC is not running, cool air won't be flowing to that room (unless you leave the fan on all the time). Since the hot room has a wall (and windows?) facing the south and east, it probably gets a lot of heat gain late in the day compared to the rest of the house. Without a thermostat of its own, it is probably going to be hot, especially in the PM, because the main thermostat has "told" the AC unit to turn off (because the thermostat has reached the set temperature).
If the circumstances permit, you could make a new hole in an external wall for a window unit. They do make "in the wall" units designed to go between 16" studs, but you can also install a stock window unit (less expensive) across 2 stud bays (30" total opening width) with the installation of a simple header board top and bottom. This gives you a unit with its own thermostat, and if the bedroom is the only room occupied at night you could save money by letting the rest of the house get warmer (e.g 80 degrees at night) and only cooling the BR down to comfortable temps. Have a setback thermostat get the house cool again before you wake up each morning. Window AC units can be noisy, but they aren't all noisy. Some people aren't bothered by them at all.
If there's a bathroom attached to the hot bedroom, open up the register in that bathroom and leave the door open to the BR.
Be sure there's a way for "return air" to leave the hot BR. This can be via a dedicated return air duct or (in more modest homes) the door is cut with a gap underneath. The gap should be large--about the same size as the "supply air" grills serving the room. If practical, leave this door open at night.
Low-cost approach to active cooling: Is there a convenient spot to run a duct or two downstairs from the hot room to a cool hallway, common area, etc? Maybe an open stud bay on an interior wall? Just bring the cool air from the lower level of the main house to hot BR using a quiet fan in each duct. Leave them on all night. Because the air coming in won't be very cool (i.e. it will be 72 deg instead of the 50 deg air an AC unit might supply) it may take quite a bit of flow to make up for the heat gain in the hot BR, but it's a simple answer. Also, this solution will provide direct sound transmission through the duct from the BR to the other end ("bridge to engine room, full ahead"), which could result in some awkward situations . . . .
Work to reduce the heat gain in the hot room. A few inches of added insulation above the ceiling, awnings/reflective film over south facing window, a shade tree etc.
Small window AC units can be bought for about $150. If you've got a suitable window, it might we worth a try. If you don;t like it, you could sell it without taking much of a hit.