You did not develop kidney stones on Keto. It takes years for most to develop and the circumstances require specific biological mechanisms such as vitamin K2 deficiency, excessive sugar consumption, oxalates consumption, and other factors.
What you experienced was the crystals you had from a previous WOE that dislodged as they broke down because Ketogenic breaks down crystalline formations.
Now, I will state this, lemons are not essential, but are medicinal and do help to break apart formations in both gout and kidney stone sufferers.
Meat does not cause uric acid to hike, sugar, alcohol, and plant proteins do. The plant proteins are coupled with massive amounts of carbohydrate, which blocks uric acid excretion. The results on purines are so inconsistent that they can't even be considered, but the results with those other factors are pretty solid. What meat does is cause a flare with temporary purine hike.
There is natural medicine for this and it's supported by science:
1/2 cup tart cherry juice
2 tbs raw apple cider vinegar
2 tbs lemon or lime juice
1/8th tsp magnesium sulfate
1/2 tsp pink or sea salt
Enough water so it doesn't kick your face in.
Tart cherries have been shown through study to break apart crystalline formations of many types, and has been medicinally recommended for a few years now. Vinegar is a healing substance that has a similar function, but other properties, one of which is diuretic, making you pee and clearing the urinary tract out. Sour citrus has polyphenols or plant nutrients which break apart crystals as well and stimulate the kidneys, which helps them recover from a high carb diet. The magnesium relaxes smooth muscles in the urinary tract and helps you pass the stones easier. The salt is if needed and balances electrolytes as the diuretics can deplete them. Water is water. You don't dilute this, it will be like drinking pure vinegar in intensity.
Once you have cleared out, you will not form stones again. The symptoms of the stones leaving is not you developing them. Meat does not and never has caused any diseases involving stones.
Edit: Retract the vitamin C on breaking stones, it helps get rid of UTI's. I need to find the studies I misplaced showing the polyphenol reaction to stone breakdown.