Just letting it run through, or otherwise cycling it really quickly and cheaply, does seem possible to me. I'd be interested to know if there are any fundamental reasons you can't let it flow, like if water moving past the electrodes decreases efficiency or something. You would at least need to watch out for the current taking away your product.

Deposits, yeah, but that alone doesn't sound like a showstopper. This is far from the first engineering application to use seawater, there are probably ideas out there that they can borrow.