Hm, LateX is not the easiest thing to move to on a deadline (I tend to recommend a Markdown workflow with Pandoc which is simpler and far more flexible).
Nevertheless, Bookends is an excellent choice for LaTex companion. Here is what I would do:
1) If you haven't done it yet go to
Settings ▸ BibTeX and enable it. The User1 field becomes the Citekey field. You can set it up so copy citation copies the citekey but perhaps LyX will do this?
2) Choose your citekey format -- Author + Year is the simplest and creates smaller citekeys (if there is more than one a letter is appended, e.g. doe2022a doe2022b), but you may prefer longer and more unique keys if you use multiple databases etc.
3) If your Citekey fields are empty you need to populate them. Select all references and
Refs ▸ Generate Citekey.
4) Use the
File ▸ Sync Linked BibTeX File option to create a one-way sync of your Bookends database to a BibTeX file. I've not used LyX but you should be able to point it to this .bib file so it can be used.
The next steps depends on if you will use BibTeX or BibLaTeX as they may use different formatting in the document. I would always prefer BibLaTeX which LyX seems to support:
https://wiki.lyx.org/BibTeX/Biblatex — as far as I can glean LyX should allow you to insert refs from the linked BibTeX file and you should be good to go...
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Conversion: if you were using Word, I assume you already had many embedded references in your documents, how are you going to convert them? It may be possible to use a search + replace script to autoconvert Bookends temporary citations, so:
{Wilberry and Schwartz, 2002}
would become
\cite{wilberry2002}
If you choose to use the author + year citekey format for example?