I agree with Phil that both are acceptable. Making the verb plural to agree with the "real" subject ('bottles') is OK; making the verb singular to agree with the dummy subject ("There") is also OK. Both sound fine to me. NB It is helpful for us to be aware of members' mother tongues : this is because we can often give you a better answer if we know what your linguistic starting point is. But if you really can't put your real native language on your profile, how about ticking the mysterious 'Other'? Then, at least, nobody is distracted by the idea of you being a native German speaker (and you don't need to explain anything).

You know English, so would you say “there isn’t more than six bottles” or “there aren’t more than six bottles”? I’d say the latter, so it’s grammatically plural, agreeing with the real subject (six bottles), not with the dummy subject, “there”. On the other hand, there is a strong tendency, when the number is not obvious, for native English speakers to make the “there is” construction singular, even when it should be plural. So, for conversational English, it really doesn’t matter much.

There doesn’t need to be more than 6 bottles There doesn’t need to be that many bottles

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There doesn't need to be more than six bottles. There doesn't need to be that many bottles.