I think I would go with keep the new product on the same site for the simple reason that new domains do take some time sometimes to get well ranked and/or trusted...

On your current site you could make a new page or new section with sub-pages as well, that relate to the product...and also link to the new product page from several 'predominant' pages on your site including your home page.

For one of my sites, launched a new section offering reviews (of my competitors services) and linked to it from a few places on my site, I'm ranked #1 for my keywords and ranked #1 for the reviews searches now too...

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