^{(I suspect this is not actually a case of double-negation — nor one of negative-concord, either. Maybe.)}

It might sound a little bit old-fashioned or formal, but yes, your second sentence is certainly allowed. See here. I might well use your sentence, at least in writing or formal, crafted speech.

It’s a pernicious myth that nor cannot be used without having been preceded by *neither. It’s up to the writer and their preferred style for that sentence, not some imagined rule grammar:

1. Those on the bottom had neither the ability nor the opportunity to get to the top.
2. Those on the bottom did not have the ability, nor the opportunity, to get to the top.
3. Those on the bottom did not have the ability or the opportunity to get to the top.
4. Those on the bottom did not have the ability or opportunity to get to the top.
5. Those on the bottom hadn’t the ability or opportunity to get to the top.
6. Those on the bottom had no ability nor opportunity to get to the top.
7. Those on the bottom had no ability to get to the top, nor the opportunity.
8. Those on the bottom had no ability to get to the top, nor had they any opportunity to do so.

I have seen broken software try to tell you that only one or two of those are allowed, but that’s just a bug in the limited experience with real English by the misdesigner of that software.

^{(I suspect this is not actually a case of double-negation — nor one of negative-concord, either. Maybe.)}

It might sound a little bit old-fashioned or formal, but yes, your second sentence is certainly allowed. See here. I might well use your sentence, at least in writing or formal, crafted speech.

It’s a pernicious myth that nor cannot be used without having been preceded by *neither. It’s up to the writer and their preferred style for that sentence, not some imagined rule grammar:

1. Those on the bottom had neither the ability nor the opportunity to get to the top.
2. Those on the bottom did not have the ability, nor the opportunity, to get to the top.
3. Those on the bottom did not have the ability or the opportunity to get to the top.
4. Those on the bottom did not have the ability or opportunity to get to the top.
5. Those on the bottom hadn’t the ability or opportunity to get to the top.
6. Those on the bottom had no ability nor opportunity to get to the top.
7. Those on the bottom had no ability to get to the top, nor the opportunity.
8. Those on the bottom had no ability to get to the top, nor had they any opportunity to do so.

I have seen broken software try to tell you that only one or two of those are allowed, but that’s just a bug in the limited experience with real English by the misdesigner of that software.