”Iute” is adverb too, see my first post, like in ”iarba crește iute” (the grass grows fast). In fact is mostly used as adverb. https://dexonline.ro/definitie/iute
”Eu manânc iute piperul” (attach it to the verb) leaves no ambiguity of its function or meaning, it means ”I eat the pepper fast”. The Duo's version is ambiguous. Without suitable context, a native Romanian won't know the exact meaning, either ”I eat the hot pepper” or ”I eat the pepper fast”. The ”antonymic” sentence is also non-ambiguous, ”Eu manânc încet piperul” or ”Eu mănânc piperul încet”, leaves no doubt, it means ”I eat the pepper slowly”, and that is because the pepper can not be slow, in spite of the fact that ”încet” is also adjective (like in retard, slow-moving, lazy, etc).
To remove the ambiguity, you have to say ”Eu mănânc piperul cel iute”, now this qualifies the adjective (I eat that hot pepper). There is not much place for a synonym, beside of the ugly construction (due to repetition) ”Eu mănânc piperul piperat” (however, grammatically correct). For the other meaning, beside of attaching the adverb to the verb (as shown above, and then there is no doubt), you also have plenty of synonyms to use, i.e. ”Eu mănânc piperul repede/imediat/etc”.
"Piperul" is often used in association with "iute" to mean "hot pepper". As a native speaker, I didn't think about "iute" used as an adverb, not even a second. I probably think that most Romanian speakers will interpret it as Duolingo's answer and not as your additional answer. Placing "iute" just after the verb eliminates any confusion as you said.