I'm not sure I understand what your trying to do , but generally using a volume control on an inexpensive DAC is not wise. Your integrated amp will need "main in, pre out" connections to achieve this.
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First - I would be less concerned with the chip than the overall sound. There is a lot more to a DAC than the chip. That aside, I would not be concerned about running a dac with a volume control into an integrated. Using the DAC at a higher volume control (3/4 full or so) will generally give a better signal to noise ratio, but the DAC should be set to a place where the integrated's volume is at a "normal" setting - not unusually high or low. Other than that, don't worry. (of course, a cheap POS volume control can degrade the sound, but that's for another day)
Preamps are the bain of high-end audio. I used to use a highly modified Mark Levinson #38 and I thought it was transparent. Not even close. Now I avoid them like the plague. Worse than digital jitter. They usually add more distortion, compression and noise than any other component.
When I need a preamp, I now use a passive transformer linestage. If they use decent transformers, these are better than any other passive or active linestage. Here are some versions:
If you do get a DAC with a preamp, make sure that the preamp/volume is really high quality or that you can completely bypass it when selecting line-out.
Going through 2 active preamps is twice as bad as one, which is bad enough by itself.
I would be careful about what you read on D/A chips. This is not a good criteria for selecting a DAC. Some of the older chips actually sound best. I wanted to use the AKM chip myself in a new DAC, but I found it to be too electronic sounding. A result of the switched capacitors I think. Good chips abound from many manufacturers, including Analog Devices, TI and Wolfson. The implementation of the DAC is more important than the D/A chip as long as they didnt pick a poor one. It is actually more important to avoid bad Digital Filtering, too many analog stages, poor op-amp design (not all bad) etc.. Low jitter of the digital source interface is actually the most important thing to look for.
Best thing is to read reviews and listen to it yourself if possible.