There are a number of (very expensive) options from Barbieri which work great for this purpose. I'm using a LFP S3 and it's great if a little finicky.
I bought a DTP-41T before I caved and got the LFP but it never seemed to work super well-- I got better results doing one patch at a time with a spectroscan-T.
The i1pro3 can do backlit measurements now-- you basically place your substrate on a backlit panel, do a flat-field compensation measurement to compensate for color/density differences across the panel, then measure the chart. I'm guessing it still has to do one patch at a time (I haven't used it myself) since it takes longer to measure transmissive patches than reflective ones, so it would probably be extremely tedious to do manually. Lastly, you can use older i1pro versions with argyllcms to measure transmissive patches, but I don't know of a way to compensate for the backlight panel, which really needs to be done unless you move each patch into the exact same spot to measure it, like you do with the spectroscan-T.
As to your last question about what light as a source-- if you're going to go the i1pro way, you should use a light as close as possible to the one you're gonna end up actually using in the final lightbox. If you go with one of the purpose built tools for this purpose, the bulb has a known spectral quality, so you can interpolate to a lot of different light sources from your measurements.
