If your library is in the US, then you could get a Kindle. Is there any reason it *must* be EPUB?
Kindle books can be checked out from OverDrive and redirected to a loan redemption page on amazon.com, where it gets added to your account and then you can sync from the Kindle.
Unfortunately, OverDrive only offers Kindle as a checkout option in the US.
Failing that, what PeterT said might work. Depends on how the OverDrive Chrome app works.
I have a nasty suspicion it would store the books in a heavily obfuscated Chrome cache. Also, you'd need to log into the same Adobe ID on the app and the ereader -- but I believe OverDrive recently took steps to make you use an OverDrive login which masks an Adobe ID???
I could be wrong -- I don't use it myself.
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