Is it possible to use THREE.Water in combination with my own texture? Or does anyone have another suggestion how I could make it look a bit more cartoony?
THREE.Water2 are not intended to use external maps. If you study the details of the implementation, you will see that both objects reflect the actual environment on the water surface in realtime.
THREE.Water2 also supports refraction so objects under the water surface are actually visible. An additional map does not fit in this concept.
If you need a more cartoonish look and feel, I suggest you use a totally different implementation which is less expensive. Just google for “toon shading water” and you will find many interesting tutorials about that topic.