Hi there
I know the game you mean, we used to play it as kids, but we would take turns and any matched pairs are kept.
I have worked with making cards, virtually shuffling cards and online dice games and communities before, so have the ideal experience to do this, but, there are some things I'd like to discuss.
My proposal bid amount is of course, outside of your budget and is only to send you a standalone HTML and Javascript page that can be uploaded to any server, where a player can have the cards shuffled and dealt (reset/start again) and then follow the process you spoke about.
I would use pictures of the cards and card backs that can be replaced easily, if you like.
There would be no login or animation, when the cards are clicked, the image of the card back would be replaced with an image of the card front and the player can continue.
Please let me know if this is something you can agree to.
If you'd like more, such as multi-player or a log-in system or a very fancy design, I am afraid I would need to alter the bid amount.
I hope to hear from you.