Using 1/4 and 3/16 together should work just fine. That's what I use on my keezer. In fact, 3/16" tubing over a 1/4" barb virtually guarantees a leak-free connection.
Are you sure your disconnect is 3/16"? All the barbed ball-lock and pin-lock fittings I've seen are 1/4". If it's threaded and you're using a swivel nut connection, it could be 3/16", but most of those are 1/4" as well. If the nipple on the shank is larger than the disconnect, it could be 5/16".
If it is 1/4", 3/16" tubing will fit over a 1/4" nipple just fine. Just heat some water in a microwave-safe mug, dip the end of the tubing for about 10 seconds to soften it, then it should slip right on. If it's 5/16" and it has a removable tailpiece, I'd swap it out for a 1/4" one. If it is 5/16 and it doesn't have a removable tailpiece (nipple is welded to the shank) I'd get a new shank.