I think you should be fine. If the CPU isn't incredibly high end, it's safe. There's one version of the GT730 that, according to the GeForce website, has a TDP of 49W, and the others have about 25W.
http://www.geforce.com/hardware/desktop-gpus/geforce-gt-730/specifications
Their 300W recommendation is assuming you have a high-end CPU, possibly multiple hard drives, etc.
I've been running an R7 250E (55W TDP) on a Dell Inspiron Slim Desktop equipped with a Haswell Pentium CPU (54W) with only a 220W power supply trouble free for nearly two years. I am cutting it somewhat close, but the 12V rail can provide up to 214 W.
I also have the same issue where the power supply is some weird shape to fit this particular case, so upgrading the PSU was probably out of the question.