I have a 6570, and It seems to be running fine with my generic 250w PSU, but my processor is lower spec than yours AMD Athlon II X2, but I have 8GB ram as opposed to four.
I also have a slim HTPC case, and the only bad thing about it, is that the selection of PSUs (TFX12V) that fit that form factor seem to only range up to 400w and don't look like very reputable brands (Athena Power, FSP Group, SPI), apart from one which is a Seasonic (http://www.newegg.com/Product/Product.aspx?Item=N82E168...) - Seasonic is very good.
A lot depends on the quality of your 250 Watt PSU, Lower end ones do NOT deliever full rated Output above 22 C, Good psus are rated upto 40 C.
6550, 6570 and 6650/70 are very close on power (didn't check 7750/70).
System at Idle = approx 144 Watts, under load (running Crysis) about 198 Watts.
Bottom Line: With a QUALITY PSU you are close to the compurt level. That is you are drawing about 80% of available. Would NOT recommend using a LOW quality PSU at 80% of capacity - Some LOW quality PSUs only deliever about 60->70% of rated capacity when under load.
One question, what do you mean by
(it has a smaller x16 slot than standard)
There is no such thing as a smaller 16 than standard, if its smaller its either 8,4,1x pci express not 16. There are 16x connectors that only support 8x or 4x speeds but they are the same length, or it could always be a simple pci connector.
This image shows a 32 bit pci slot and 1x 4x and 16x pci express connector sizes for your reference, I read a short review on your desktop and it states that only a single x1 connector is free on the board, but I cant say for certain. http://upload.wikimedia.org/wikipedia/commons/f/fc/PCIE...
Your not screwed yet however, you cant use some of the higher end cards but you can still upgrade your graphics, however I recommend you get at least a 3-350 watt psu first.
best available at newegg for a 1x slot, if you need low profile for that case it comes with the bracket you just have to pull off the second connector and swap it out.