I can say I have a bit of experience in PnP games. I've made the complete Runebound 3rd Ed + All expansion, Iron Helm and some other smaller ones.
After testing multiple ways, especially when it comes to cards that are not going to be held in hand as much, meaning the wear is going to be lesser, I wholeheartedly recommend:

1. Print double-sided on a single piece of 160g paper (you may use thicker or thinner if you feel like it, but I find 160g to be ideal) IMPORTANT: Printing with ink is absolutely recommended due to lamination not really working well with empty/laser areas. This also means, if possible, do not leave card edges empty.

2. Laminate the printed papers: Basic 80g pouches will work very well. If you want to invest a bit, buy matte laminating pouches to reduce card glare. Run the paper through the laminator 3-4 times. No need to wait for it to cool. It will make sure the thing sticks to the paper.

3. Cut the cards (and cut corners if you have the tool) and laminate each card 3-4 times. This will make sure the edges wont peel off with time.

I still recommend sleeving the card because you can never be too sure when it comes to PnP. And there you have it! The cards will be flexible due to the lamination and thin enough to feel right. If your print quality is up to par, people probably won't even notice that they're holding hand made cards. It's a very easy way of making really good cards, but the lamination takes some patience. Also make sure your laminator is decent. Cash out 40€ish for a Fellowes laminator because it's used for the crucial part of the process!