My immediate thought is to have a 'universal review code' on your invoice, which somehow encodes the monetary amount of business as well as the date. Reviewers could use this code when leaving a review on a 3rd party site to prove they were a genuine customer. There's an opportunity for stalking or harassment if the business was able to correlate the code to an individual though, so the money and time might have to be bucketed. If a customer got a refund the same system could be updated to show the fact.

A company that sells a POS system (like Square) could do this. They'd have a record of verified purchases, and if you input your email, similar to with Amazon purchases, they can follow up with you to ask if you want to leave a review.

We were thinking about giving cards with the Tripadvisor name/link to restaurants to give then only to those customers that seems specially happy; however the local restaurants where I live are really technophobic so they just saw it as overhead with no benefit.

Technophobic? The industry that keeps insisting on just uploading PDFs of their menus instead of maintaining a proper website?

No way.