Do you actually need a DAC section or are you just saying that because it’s an option for volume control?
I’m not sure which Beresford you’re talking about, the Caiman or new Bushmaster? The new Bushmaster does have a volume control but this only controls the headphone output so the Mk 1 version is not suitable as a pre. The Caiman however can be used as a pre. I won the Caiman and it is a very good DAC for the money, that said I used it as a short term solution as a pre-amp and for me it wasn’t up to the task, it works yes but I found the sound a bit flat. Although you are working to a tight budget so it may be a practical option and a sound you can compromise with.
If you don’t actually need the DAC section and don’t need many inputs you could also consider using a headphone amp as a pre, this was suggested to me when I was searching for a pre-amp and it actually seemed like a really good idea although I didn’t go down this route. You tend to get quite a lot for your money, the can have high quality volume pots and often a good active tube stage too, have a look at the schiit headphone amps if this is a viable option.
Lastly if you do need a DAC stage also consider a pre-amp that has a DAC (rather than the other way around, ok ultimately it’s the same thing but much more attention will be given to the pre stage rather than it being an afterthought / secondary function as it is on many DACs). I reckon if you don’t mind going the 2nd hand route you may well find one of the Cyrus pre-amps with digital inputs within you price range.