I think that the comma before “with” is wrong, and “estimates” is wrong, as well. This is “methodologies”, which is in the plural form, so it can’t be “estimates”.

I would write the phrases this way:

Offered a way to close the chasm between science-led precaution and cost-benefit concerns with methodologies that raised impact costs estimated by order of magnitude compared to most previous aggregated impact costing efforts, (COMMA) SUBJECT VERB (OBJECT/SUBJECT COMPLEMENT) (whatever that needs to be added).

Now the phrases make sense even though the subject and the verb are yet to be added.

In VERY simple words, the phrases mean:

Offered a way to close the chasm with methodologies, they accepted the request/deal.

I just made up the independent clause so that you understand.

“Offered a way...” is a participle phrase, which means that it acts as an adjective modifying something.

“With methodologies” is just an adverbial phrase saying HOW they want to close the chasm.

“that raised impact costs” is a defining adjective clause defining WHICH methodologies are going to be used to close the chasm.

You now need an independent clause after those phrases.

I hope that makes sense.