I recommend separates.
I’ve been doing streaming by about every possible way in the last twenty years. Typically the streamer and DAC should be of about equal investment. You want as high a quality you can get in each (picking the very best you can). Particularly in the DAC you want it to support your musical preferences. Pay particular to the sonic characteristics. Particularly in this price category you can get highly detailed fatiguing DACs.
I recommend Aurrender streamers… used would be required since their lowers price models are over your budget. I am not familiar enough with current DACs in this price range to specify a DAC. But… you might leverage the sound quality here as well by getting a DAC less than five years old. It is likely to put you at a higher level of performance.
Typically SP/DiIF / AES are the best performing connection… but depending on the streamer / DAC USB might. But the difference is not likely to be large. Something you can play with after you get them.