It sounds like the flash on low power is a great idea to get started with. However it's probably not going to be your best choice when you start burning and dodging your prints to darken or lighten specific areas relative to the overall exposure. You usually don't want sharp lines that you would get with flash stopping the motion of a dodging flag.
In a reflective room, you could set up a desk lamp with a relatively low wattage incandescent lamp on a dimmer. A dimmable LED could also be a great choice. It really doesn't need to be that close to the print, just somewhere that you can get a nice even light across the neg and paper.