I want to have inventory that syncs between eBay?

Hi all!

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I searched a bunch but can't seem to find something to satisfy my specific scenario. I am building an online video game store, and I want to be able to go into some sort of backend and list it as available on my site, and while I do that it'll make a listing on eBay. If it sells on eBay, it'll automatically remove it from the site so I don't double sell it. Perhaps it can link to Facebook too?

Simpler:
I have 1 of Game Y
I put Game Y on my site for sale with an inventory of 1.
Game Y is automatically listed on eBay for the price listed on my site.
Game Y is sold on eBay
Game Y is set to inventory level 0, thus being removed from view on my site.

Which platform is good for this? I've been looking at SellBrite, WooCommerce, BigCommerce, Shopify etc.

I plan to have things that are website only, but basically any of the platforms can do that easily so not too worried about that.