Here's the lowdown.
Do a bit more research is my advice but here's my further advice:
First, the CPU.
Per core, the phenom II is faster and games will make use of the L2 cache, however, at the pricerange for graphics you are looking at, this doesn't make any difference because your CPU will be slowly walking in order to keep up with a 5450. Even An Athlon II x2 at 2.8GHz would be fine for anything up (pushing it a bit) a radeon 5770.
BFBC2 is a bit of demanding game and makes great use of multiple cores.
MW uses only 2 cores. MW2 can use 3 cores.
Taking this into consideration, a quad core will likely benefit you more in the long run, even if it is lacking the L3 cache and is 0.2GHz slower. BFBC2 is a bit of a more CPU demanding game than other games.
The video card:
I'm just going to come out and say it. The 5450 is a poor choice and you obviously did no research into video cards. I have a 4850 (equal to a 5750) and I can't play BFBC2 at 1920x1080 unless I turn down all the other settings. And even then, I would be lucky to get 60FPS.
Right from toms performance charts:
http://www.tomshardware.com/charts/2010-gaming-graphics-charts-high-quality/Battlefield-Bad-Company-2,2465.html
Notice anything? That's right. The 5450 isn't even on the list because it performs worse than the 5550 at the bottom of the list.
The first modern warefare game is a bit more forgiving and I have more experience with it on multiple systems and cards. For comfortable gameplay at that resolution, I would recommend at least a radeon 5670.
So now my advice: You mentioned your CPU budget was $100. I recommend you take that and cut it in half. Take that $75 and buy an Athlon IIx3. Take the other $75 and add it to your (I assume based on 5450 prices) $50 graphics card budget. That should be enough to get a radeon 5670 or a 5750.
http://www.newegg.com/Product/Product.aspx?Item=N82E16819103872&cm_re=athlon_II_x3-_-19-103-872-_-Product
http://www.newegg.com/Product/Product.aspx?Item=N82E16814161309