You can make your fishing rod invisible anytime you want, tho it does require you to lose one bait to do so.

1) Cast your rod.

2) Pull it back in early

3) Immediately recast your rod

The animation will go through the reel in and recast, however your rod will then disappear and your left holding an invisible one that still works.

Also, the "used bait" item does nothing, it says it's a use-able bait in the description, however it's not selectable as bait.