I would make some distinctions here: a lot of the laws in Saudi Arabia are repressive, but the government (read: the royal family) isn't especially religious. Rather, the government "outsources" some of the lawmaking to the clerics, whose fatwas may be enforced, not enforced, or partially enforced. The government has the final say.

Further, these kinds of rulings don't apply to non-Muslims, so nothing is violated by inviting people from around the world (non-Muslims) to play chess in Saudi Arabia (as long as they don't visit the Hejaz).

Another thing to note is that Saudis play video games and watch foreign (read: American) movies. It would be pretty outlandish if that was tolerated but chess was banned. So I highly, highly doubt that chess is banned.

Straying off topic a bit: the dynamic with Israel is interesting. The Saudi government is pretty much aligned with Israel (anti-Iran, close American ally, anti-Islamist uprisings, wish to maintain the status quo), but they have to keep up a facade of zero diplomatic contact because being anti-Israeli/pro-Palestine is part of the Arab-Muslim identity at this point. You can bet Israel and Saudi Arabia have contact, it's just not out in the open.

Finally, something interesting to note is that Israeli-Arabs can visit Saudi Arabia and often do (for the Hajj). They just have to do get a Cypress or Jordanian visa and cross through Jordan.