What is a good substitute for mayo and miracle whip

Not sure this would work for your recipe, but as someone who doesn't like mayo or miracle whip, I've had good luck using greek yogurt in place of mayo.

If he's not allergic to milk, try a milk mayonnaise:
http://leitesculinaria.com/32983/portuguese-milk-mayonnaise.html

Or you can try Just Mayo which is vegan, using a pea puree of sorts and oil. I've tried it and it's pretty good. And No, I am not an employee nor do I have family invested or working for Hampton Creek :)

i will echo nancy's question... is he allergic to the eggs, the oil used or something else? Even so, a substitute would be a fat that he'd tolerate, perhaps use mashed avocado ?

Make an aioli

Mayo is mostly oil emulsified into egg yolks. Which is he allergic to, if not all?
If the eggs, use plain good olive oil in making the tomato-bacon cups. Otherwise, try some other fat he can tolerate and will taste good with the tomato-bacon recipe.

Depending on his allergies, you might be able to try a vegan mayo, or you could skip the mayo on his cups.