*The SLIM GTX750Ti is your best bet. More info:
You may run an HD7770 with that power supply, though you would need a MOLEX->PCIe 6-pin adapter likely (some HD7750/70 models have it included).
No guarantee you won't have power issues though.
The GTX750Ti is by far your best choice:http://www.galaxytech.com/__EN_GB__/Product2/ProductDetail?proID=517
This is a LOW PROFILE card so should fit, and can work with most 300W supplies. It will however cost more than your budget since it's $160 but I simply can't recommend anything else with your PSU and wish to run Dayz.
If you can't stretch your budget I suggest finding a slim HD7750 though they may cost at least $100. The GTX750Ti uses the new Maxwell GPU so it's very power efficient. It's the most POWERFUL graphics card a 300W Power Supply can run.
Here's a review: http://alienbabeltech.com/main/overclocking-galaxys-low-profile-gtx-750-ti-without-limits/2/
Store ($160 USD; probably not cheaper anywhere): http://store.galaxytechus.com/GALAXY-GeForce-GTX-750-Ti-GC-Slim-2-GB-GDDR5-PCI-Express-30-DVIHDMIVGA-Graphics-CardbrbrFree-Gift-150-In-Game-for-Warface-Path-of-Exile-and-Heroes-of-Newerth_p_90.html