Build a corpus for machine translation
What you would typically do in your case is to apply a sentence alignment tool. Some popular options for that are:

hunalign: a classical tool that relies on a bilingual dictionary.
bleualign: it aligns based on the BLEU score similarity
vecalign: it is based on sentence embeddings, like LASER's.

I suggest you take a look at the preprocessing applied for the ParaCrawl corpus. In the article you can find an overview of the most popular methods for each processing step.
A different option altogether, as you suggest, is to translate at the document level. However, most NMT models are constrained in the length of the input text they accept, so if you go for document-level translation, you must ensure that your NMT system can handle such very long inputs. An example of NMT system that can be used for document-level NMT out of the box is Marian NMT with its gradient-checkpointing feature.