I don't believe there is a way to make the images process as they come into C1 Pro (except for hiring a teenager to hit the process button each time an image appears). There are settings to make your images come into C1 pro with adjustments you've made to a previous image.
The quickproof process setting will process much more quickly than the regular processing. But you will still need to select a group of images and hit the process buttom several times throughout the day. You can also choose a colorspace like sRGB in the Quickproof, but leave you HiRes images as Adobe1998 if you like.
It is unfortunate that you cannot use the images in the "Previews" folder of your session. With the Leaf software the screen previews are available to just grab right out of a preview folder like you are wanting to do. The Phase software is using some sort of proprietary file for the screen preview. You could ask Phase One about the possibility of changing this.
Phase One does have a utility from what I have been told when I asked them about this a little while ago that can extract the previews right out of the raw files. A Phase Rep told me that it should be a download off of their site. It is called Preview Convertor and it is a new and very fast utility for exporting JPEG preview files from Capture One on a MAC.
It generates JPEG preview files approx 40 times faster than using QuickProof processing.
I will look on their site to see if there is a link and then post it.