Yep I don't give mine hide chews anymore, for much the same reason (runny tum) also some of the treats you can get can be full of additives, I found the marrowbone biscuits (the ones that look like sausage rolls) can upset my boys tums,
My boys are 2years and 1year Fish 4 dog Sea Biscuit tiddler treats usually go down well with Minstrel & Branston with no nasty consequences, and Minstrel likes their Salmon Strips. Their sea jerky ones are not so good for littlies, I have found, as they have a very abrasive composition designed for cleaning the teeth, if puppy swallows them whole as Branston is likely to do, it can make them uncomfortable.
A couple of Burns Kelties before they go to bed, stops the early morning pukes.
Also check the packs of treats, some are not suitable for pups under a certain age