Are you looking to game exclusively on your phone, or do you want to hook it up to a console/controller?
If the former, consider a
Motorola Nexus 6. It has a 3220mAH battery and a large 5.96" screen. It runs for about $220 to $300 on
Amazon. However, I would only recommend this if your phone's primary function is gaming. Since Google has moved on to the Pixel, security updates have ceased and the Nexus 6 is not very well-supported. If you don't like the sound of this, here's
a list of similar phones to check out.
Any newer Samsung phone will also handle almost every app just fine. I suggest the
Samsung Galaxy S7, which is an old flagship that still holds up well. It has a 5.1" 1440p screen, 3000mAH and 4GB RAM. You can buy refurbs on
Amazon or from
Samsung, though Samsung is out of stock for the S7 right now. Certified refurbs are going for $220 to $380 right now, so well within your budget.
If you're looking for a larger battery, I was very satisfied with the
S7 Edge, which has a 3600mAH battery, is 5.5" and comes with a handy Edge panel. Please note, however, that colours might look off if you tilt the screen too much, so if you're very picky about cinematic colours, see if you can find a physical model to test first. The S7 Edge runs from $320 to $500 on
Amazon.
If the latter, look at a
Sony Xperia Z3. It has
native Dualshock support, has decent + appropriate specs (
3100mAH battery, 5.2" screen, 10-finger multitouch) and goes for dirt cheap on Amazon,
ranging from $150 to $215. Please note that it may be difficult to find one with a valid warranty, however, since it's no longer being manufactured.