amd overestimates psu requirement to make room for generic, low end psus which do not deliver enough power efficiently. it depends on your psu. what is your psu model? especially, amps rated on the +12v rail(s).
the detailed psu specs would be printed on the psu itself. if the pc isn't under warranty, try checking inside the case. however, if you put in a discreet card, you will have to open up the case anyway, possibly voiding warranty.
afaik, there aren't any new card that use as little power as the 6450. it should run on the 300w psu. my suggestion to upgrade the psu still stands.
oems use psus that support the components in the pc, not upgrades. they usually cut as many corners as possible, especially on entry level pcs. psu is usually where they cheap out the most.
I just spent a great deal of time looking at inexpensive power supplies in the 400-500 watt range, including reading (and rereading) what I think was just about every word of every review for such at NewEgg. It is my conclusion that the best power supply for the money (and it is actually just a bit less expensive than those you suggested) is the Antec vp-450: