A little update
To double check my work, after driving about 20 miles with the new fluid, I opened the fill plug to see if there were any leaks or to check if my fluid level dropped. The fluid was lower than before. I know this because I was expecting some dripping since I closed the plug while it was still dripping. I put more fluid in the pump and filled the transmission and about .2 liters went in. This had me thinking and I found that it's possible when I drained the transmission, the residual oil from the cooler had come out with oil from the pan. This was a bit confusing since the TIS manual calls for running the pump to remove the residual oil. However, I think I let the oil drain so long that the cooler had emptied.
It appears after filling the transmission with new fluid, you need to close the fill plug and start the engine and engage a gear to get the pump running. Afterwards, reopen the fill plug and recheck the oil level and fill if needed.
Since this is my first transmission fluid change on the M3, I didn't expect to do this since my previous cars didn't have a transmission cooler and pump. However, this will definitely prevent low fluid levels in the transmission.