1) moving actual product.com website to product.company.com or 2) build and let grow company.com and when it's completely indexed redirect traffic from old website to sub-domain.
product.com isn't an e-commerce and no great "click-to-sell" conversions are made within it: it's just a showcase site.
Since you say there isn't much to product.com, it doesn't deserve a sub domain or even its domain. You're better off making a sub-folder so your URLs about the products can begin like this:
Then if you want to share details about a specific product, then put a file in the sub-folder so your URL can become something like this:
I'd only consider a sub-domain if there's a massive amount of content relevant to the topic in the sub-domain itself. Since your wanted sub-domain name is products, I'd consider making a fully functional e-commerce site first before putting product information in a sub-domain.
The problem with your idea of....
build and let grow company.com and when it's completely indexed
redirect traffic from old website to sub-domain.
...is that you're putting mega strain on search engines because you're telling them to record all of your URLs for the site at one location then when the whole site is indexed, then the search engines will slow your server down because when they try accessing the URLs you provided to them at first, they will receive HTTP headers asking them to access new URLs.
This in turn causes search engines to not only make one request to every single URL, but TWO requests. The more requests there are made to your server in one time, the longer it takes for your server to process each one.
Since company.com is your primary site, you're better off keeping it, and not moving content, but if you must, then just move it to a new domain, not a sub-domain.
Also, if you move your site, you're better off making a copy of the website to your new domain and making the new copy not yet indexable, then make changes to the new copy until it is perfect, then setup redirects from the old site to the new site and make the new site indexable again.
The advantage to this is there may be fewer URLs to redirect (especially if you're adding pages in the future), AND the search engines won't have to try to index everything twice.