So when the Diddy does his dash, you can just shield grab him to punish him. Throw him forward and use his bananas against him.
Normally I would agree, but Diddy's dash attack is pretty unique, and he can perform a few attacks almost directly out of his dash attack (Usmash, Utilt) - if you try to shield grab you might get hit in the face. I'm not saying a shield grab won't work, but it could be risky.
I think Snake's tilts probably outprioritize Diddy's dash attack, or a grenade drop / roll would be a good option too.
To add on to what you said earlier, which I thought was good advice: Diddy's bananas allow him a certain control of the stage, somewhat like Snake's mines / nades and so on, but not nearly to the same degree. The difference is that Snake can use Diddy's bananas against him, while Diddy can't use most of Snake's arsenal against him (only badly used grenades). Snake has one of the best dash attacks for picking up items, and added to that you can perform the mortar slide to pick up Diddy's bananas too. You can use the bananas to hit him, or put them near something explosive so that Diddy will be more reluctant to try to retreive them / if he does you can try to punish.