Depends on which brix you're looking at. The ones with room for a 2.5" might be a bit noisy for a living room box.
What sort of RAM do you have already? desktop or laptop stuff?
You could do this:Antec ISK 110
$135 (pricey, but low TDP, great for a small case like this)ASRock H97M-ITX/ac
That gets you to $310. The ISK 110 has room for two 2.5" drives and if you've got desktop RAM it'll fit in there. Plus the i3-4130T comes with a low profile heatsink that'll fit in the case.
You've got HDMI, USB 3.0, 802.11ac wifi, and optical audio out, too, if you want it.
I feel the need to point out that a regular i3 will work too and could save you $10 / get you a faster CPU and onboard GPU, but I'm not sure of the height of the stock heatsink that comes with a regular i3. If it'll fit, that's also an option. The extra power draw wouldn't matter - the ~80W PSU in the ISK 110 would be able to handle it just fine (I ran an i3-530, motherboard, ram, SSD and 2.5" HDD off of one of Antec's 65W PSUs for a couple of years without issue, and it was a "casual use" type of system, too).