The Zoom H1 XLR will meet your battery power requirements. I recently tested it with a pair of ordinary supermarket alkaline batteries. In other words, they are unlikely to have unusual capacity. The continuous recording from a Superlux S502 stereo mic, phantom power on, ran for 4 hours and 6 minutes. It created two files of almost 1 hour 34 mins each, and one file of 59 mins. But the same batteries had also done some previous tests, which perhaps accounted to 20 minutes of intermittent use. Of course with more fancy batteries you'd get longer time.
Is it the best recorder within your budget? No. Would it fulfill your requirements? It could, but I don't know what they are. These days when comparing recorders, it seems to me that the input frequency response is essentially flat for all of them. For the majority of real world uses, the self noise of the mic inputs of most 32 bit float recorders is unlikely to be a problem. (Note that I am not conflating 32 bit float A/D converters with low noise preamps - I'm just referring to a class of devices). I am assuming the use of condenser mics with reasonable output levels. So the decision comes down to the feature set, reliability, and whether you want to spend all your budget, or save a bit. And also, whether you simply want the best, whether you really need it or not!
So if you care to expand on your requirements, you might get more targeted advice. However, the most likely contenders for the title 'best' have been mentioned in the previous posts.