Not sure why you think that. I actually do eat burgers, on the rare occasion I eat a burger at all, with no sauce, which I think is fine if the burger itself is not overcooked and made with sufficiently good meat. The patty should be plenty juicy.
A 1 tbsp serving of both will set you back 4g of sugar. Sure, they each have "high fructose corn syrup" on the ingredient list, but it's a small enough amount to have no nutritional impact. In reality, the bulk of the sugar in the ketchup is coming from the tomatoes and the mayo doesn't have enough to even warrant listing measurable calorie content from sugar at all.
Pickles, lettuce, tomato, a whole fried egg. All of these are options that don't include sugar. There are options outside of ketchup (corn syrup) and mayo (probably corn syrup). Except the pickles. I bet those have sugar.
But I'll fight anyone who eats hamburgers without pickles.