On average there is about a 11 watt(20 watts if you want to run furmarks) difference from a 5750 to a 5770. So yeah, if you are that close to the power supplies limits, either card is bad.
The 5770 is the only card that makes sense to use for you.
I am not going to lie, I have not used your power supply(thus the
being extremely overcautious) and do not like its older design(as it is at best like a modern 300-350 watt unit anyway, so i would consider about 250-300 being its top end power output. Do not be fooled by overall power, its all[ok, mostly] about 12 volt these days).
Personally i am running a 5770 + i5 750 on a 300 watt power supply without issues at all(and never pulling over 200 from the wall(Prime95 + Games) most games are in the 140-150 range. The idle is under 40watts)
Can i make a promise on a power supply i have never used? Nope.
If you are worried, or want to know just how much power the system takes, pick up an energy meter(it may surprise you), it will tell you how much power you are pulling and DC power is lower since there is a loss swapping AC to DC. Just don't pull much over 250 from that thing and you should be fine.
I use this
http://www.upm-marketing.com/products/em100.html