Yeah, that's the classic trade-off. What saved me was mixing fibers. I started blending the recycled pulp with a bit of new, long-fiber stuff like cotton or hemp. Even a small amount makes it way stronger without really hurting the eco goal. You can also try pressing it longer to really get those fibers to lock together.
Ever notice how that mixing trick pops up everywhere? Like in baking, where a little vinegar can make dough rise better without changing the taste much (same idea as your fiber fix). It's all about those smart little blends.