If the grips are rubbery, the streamers should push in, though some force will be needed. That tight fit is necessary as there's only friction to stop them falling out again.
I certainly wouldn't cut the grips, but if you're really struggling, shaving the end of the plastic plug so it's more tapered will be all you need to do as once it's started you can push quite hard. That assumes the grips have a hole at all, as pictured. Most do on kids' bikes but some don't (e.g. the ones I put on my daughter's bike that came off my mtb)
It's a few years since I fitted streamers, but I didn't have my full tools with me at the time, and didn't have too much trouble.