I still won't take my children up due to my lack of experience, but I hope to able to as my experience grows. I will have to convince the Ex Mrs. tacpot that I am safe at that point. My children are also likely to be teenagers by then, and so a little more aware of the risks that might be involved and be able to judge for themselves.
So I think Mrs. Wombat is just being vary cautious by insisting that two pilots are on board. After all two pilots means twice the eyeballs looking out, twice the brain power if things go wrong, and she perhaps understands that one adult may need to supervise the children, leaving you free to concentrate on the flying. Fuji Abound's solution seems the best. Find a friend who will come with you for free.
Instructors and Examiners have to earn a living, but I don't see the need to pay them to sit next to you as safety pilot, when another PPL would do. As Pilotage and Fuji say, once you have flown with other people for a while (both pilots and non-pilots) and they have reported back to Mrs. Wombat that you are very safe, I'm sure some of her resistance will disappear.
An article in the PFA magazine recently described a similar situation, where as the aeroplane took off, Mum on the ground thought "My whole life is in that aircraft!" - so you have got to see Mrs. Wombat's point, she doesn't want to lose any of you!