You're really going to want a spindle with a 1/2" capacity. I doubt you'll find anything with a 3/8" shank that's larger than 3/8" diameter. Your best option might be to use the largest straight 2 flute router bit you can find with a 1/4" shank.
With a 1/2" shank, I'd recommend these, but you're going to have a very difficult time with that small collet.
Mini Insert Spoilboard Surfacing & Rabbeting Router Bit with Scorer 2 + 2 Design -Toolstoday.com- Industrial Quality
Spoilboard Surfacing Insert Cutter Router Bit - Toolstoday.com - Industrial Quality Insert Bits