Thanks for all the input guys. The passport stamp is not my concern at all - I've been to Israel before and have always requested no stamp. As Joshwex says they don't give you an additional piece of paper - they store the information electronically (and for posterity).

So RJ sounds like an easy issue - and it seem that ex TLV it might be simpler. So then the only concern will be how check-in staff for BA in Saudi might react?