Any motor oil would be fine. You would not want to use an oil like 3in1 or sewing machine oil as it would be too thin. However, since you already have the Reese lithium based grease I would put a dab on the top and bottom of the trunnions and call it good. Also make sure you use it on the ball too! Actually, any good wheel bearing or other grease from a grease gun would work fine. Just don't run it bare!
Barney
2004 Sunnybrook Titan 30FKS TT
Hensley "Arrow" 1400# hitch (Sold)
Not towing now.
Former tow vehicles were 2016 Ram 2500 CTD, 2002 Ford F250, 7.3 PSD, 1997 Ram 2500 5.9 gas engine