Hi mate, I have the "conexant" drivers as well. For some reason, they don't work for me too with Asio4all. In Reaper, devices, per default asio4all has the "digital out" on, and while I go into the asio4all preferences I for the hell of it cannot enable the "normal outputs", as I don't need the digital out.
So I get no sound out of Reaper too with the bloody conexant inbuilt card (which, to be frank, is a bit crappy anyways.)
My solution was to get a usb soundcard, in my case the old Edirol UA25, which should be available second hand if you'd like to have this or a similar one, as Edirol produces another card now, UA 25 EX or whatever it's called. The card is not loved by all, but runs super-stable, and with it Reaper just works 100% like with all asio-soundcards
Sadly enough I can only see what the specs tell me about your ART USB dual pre. I use a Presonus tubepre, which has NO built in soundcard like your ART has.
The "it works" sentences from ART just mean that you don't need to worry you had to find some other files to install your ART. It does not say if there are asio drivers.
And as the ART has no Asio drivers, and the conexant card in my case at lesat does not work right (this forum, if there is any, will produce a solution for us anyway, pretty soon, he he, if there is one) - the problem seems not to be Reaper-related.
For a test - but with bad latency - you could indeed choose the other possibilities in Reaper, but as already said - Asio is by far the best... I can, if my Edirol is not near, get Reaper to play stuff with direct sound and other settings, but this is of course nothing for recording or for to work with great success ;-).
So if not someone would come with another solution, as far as I understand it you would need a nice usb soundcard, if you use a laptop, or a nice card for your Desktop. Very cheap and great are for example the m-audio 2496 which I use, or some emu cards, but there are lots available now.
If that is a no-go for you, getting another soundcard - then excuse my long post
. And I am nearly sure I misunderstood this and that, not knowing your equipment well. Well.... others may chime in maybe... All the best!