"I put in a lot of butter."
While I do think that the default translation is the more natural one I do agree that your version could be accepted. Unfortunately, it sounds more like 'many butters' than 'a lot of butter'.
I agree there is a difference between the two but I don't think you are right. たくさんの can be used for both "many" or "much" (like たくさんの雨が降りました、たくさんのお金を儲けました)
The only difference is that the default version focuses on the たくさんの part, like バターを「少しではなく」たくさん入れました, while the new version focuses on the バター part, as if to say たくさんの「牛乳やマーガリンではなく」バターを入れました
I'm not sure if I correctly expressed what I think, but anyways, I think both versions are okay.