A colleague of mine was given a room full of students so bad they literally vandalized the classroom. She was on the elderly and frail side but it was hardly her fault. They needed a big hefty macho male teacher to get them going in the right direction. It ended up badly for her; she lost her job. Shame on the administration.
"I've only..." is fine in English, even if you'd not use it personally.
Indeed, it's rather a little-used form, and probably is only in the database because DL sees all contractions as valid (so for example, if "I have only..." is accepted, then "I've only..." will be accepted automatically too).
In many cases, such as this, that's fine, even a little unnatural to most. In some others, such as, say "How many pencils do you've" (from "...do you have"), it means a grammatically incorrect sentence gets accepted.
Your version should of course also be accepted; do report it if you get it again / haven't already.
“I only have the bad students.” — Should this be an accepted translation of this phrase? I mean, look at it this way: According to the teacher, there are ‘good’ students, who spend the lessons doing school work and suceeding in doing so, and there are ‘bad’ students, who only spend the lessons chatting with each other and not doing any school work at all. A teacher, somewhat annoyingly, says ‘I only have the bad students (while all the other teachers go around teaching the good students’. Would it make sense to use the definite article in that case, when referring to a specific group of ‘bad’ students?
I notice that the verb есть is omitted from what would normally be the phrase У меня есть. So far, if my memory is correct, this has occurred only where the sentence includes a negative (нет). Does any adverb which modifies or qualifies the statement 'X has/have', such as только here, make it possible or even mandatory to drop the verb?
I don't know if adverbs make it possible to drop есть, but I got this from the notes of Basics 2:
Omit ”есть” if the existence of the object is obvious or not the point — very typical for describing traits or a number of objects (“Tom has a beautiful smile/large eyes”, “She has a very fat cat”). This also applies to expressing temporary states and illnesses (“She has a migraine”).