Massive input delay during boss fights

Here's a few that could potentially resolve your issue.

First off, set Lag Compensation to None (Settings -> Lag Compensation -> None). Lag Compensations places a delay on your inputs after predicting what you're going to do. You don't need lag compensation on low ping servers and especially not when you host your own world. Will it address the large input lag spikes? I would say Lag Compensation addresses general lag and the 2nd point more so your lag spikes.

Secondly, what mods are you using, if any? Some mods are notorious for lag, (I'm pointing my fingers at you, Display Food Values). The more mods you have on, the more you risk in-game issues or lag that occurs because those mods exist. Disable all your mods and then re-enable them individually to see what's causing the lag. 

There could also be an issue with your graphics card, though I highly doubt it and can only give advice if its an Nvidia graphics card.

If it's Nvidia, right click on the desktop, go to Nvidia Control Panel -> Manage 3D Settings -> Program Settings -> Don't Starve(dontstarve_steam.exe). Choose the following settings.

Select the preferred graphics processor: High performance NVIDIA processor

Power management mode: Prefer maximum performance

Threaded optimization: ON

Triple buffering: Off

Vertical Sync: Off