Wanted to check back in and show what I came up with. I ended up going to the hardware store and walking the aisles to figure out a way to make this a bit more "elegant."
So here's what I came up with.
I used a 3/8" compression fitting to 1/4" NPT female coupling. Then I bought a 1/4" Nylon hose barb. The 3/8 Compression Fitting threads into the stout faucet tip perfectly. Then the hose barb threads into the other end. A short piece of tubing attached to the hose barb and forced over the top of the picnic tap and...viola! It works awesome.
Picture posted below. The beer is a Ordinary Bitter carbonated at 1.5 volumes of CO2, so very low carbonation. I have my regulator set at 17psi, but I use some of those "mixer stick" things that were posted on here a while back to reduce the pressure in the lines allowing for much shorter hose runs. Anyway, here's some picks of the whole contraption and a pour as well.