Austen modestly and famously decribed her art as 'the little bit(two inches wide) of ivory on which I work with so fine a brush, as produces little effect after much labour', but her novels are suffused with greater ambitions.

Hello, I have a question with the second 'as' in this sentence - ", as produces little effect after much labour". Did author use 'as' here as an indication of reason? If otherwise, can you explain to me what role it is playing? And I'd appreciate if you could offer me some other examples with this kind of structure!

The "as produces" basically means "yet resulting in" or "but resulting in".
You can thing of "as" as "that" sort of, but there is a nuance of arrival at a result I think.

It refers back to the extent of action performed in the first phrase and points towards the result of that action.

I ate so heartily as finally satisfied my hunger.
I drove so far as finally I reached the coast of California.
People showed up in numbers as were to be expected.

I must say it is is a difficult structure to explain or even generate examples for. I couldn't find an exact reference googling, but it is definitely correct.

This is something you will come across in advanced reading mostly I would say.