Risky. Your GT 630 is probably only 5-6" long and the HD 6790 is over 9" long. Whether it will fit under the heatsink housing is questionable. Also, it will not work with your current power supply as it requires additional power from a PCI Express 6-pin power cable which you do not have. The HD 6790 is a rare find anymore with the new HD 7790 out. The HD 7790 uses a lot less power and there are many more options as far as what might fit properly to work. However, you still need the PCIe power cable. Your options are to get the HD 7750 as suggested or upgrade your PSU and get the right HD 7790. For example, this HD 7790 HERE is only 6.6" long. You could look at some of the HD 7770s too and with a PSU upgrade you'll still be within your budget.
Note: Now that I look at your case design, I don't believe that you can use a dual slot card and only single slot cards will work. You simply do not have two slot covers available because the cards face up instead of down like in a normal ATX case. If the cards faced down, then you would have no problem at all. There are some single slot HD 7770s, but I see no single slot HD 7790s.