There's absolutely no difference, beyond initial water weight loss. That goes for any diet - the same deficit will result in the same weight loss, regardless. The thing that matters most is that whatever way of eating ('diet') you choose, is that it's sustainable for you. Many people find keto or low carb more satiating, and that's why it works for them (and obviously some people do it for medical reasons). That's not universal though.

If you want to ignore medical advice on food intolerances, that's up to you. Just because you don't notice any difference though, doesn't mean there isn't internal damage happening when you eat those things.

No worries, I know how hard it can be with intolerances, especially when the effects aren't immediately obvious!

We all have different experiences and mine is as follows:

I lost my excess pounds in 2013-2014 averaging 128 total carbs per day. At the time I knew nothing of total carbs, net carbs, keto, etc yet I lost 63 pounds in 53 weeks. I just ate less than I would like to eat and eating very limited amounts of things like cheetos, fritos, doritos, cookies, cakes, pies, etc. I love that junk. I gained weight eating that tasty stuff. The sweet/fatty/salty highly palatable foods specifically engineered to cause (some) people to overeat them. I digress....

In 2015 I trialed eating ketogenic for reason other than weight management (cervical dystonia a neurological movement disorder). While it seemed effective for my dystonia initially the effect was not long lasting. I ate what most would consider a Medically Therapeutic Ketogenic Diet for 1 full year. It was very high fat. Both protein and carbs were limited. In regards to weight loss/management....I intentionally maintained by eating the proper amount of calories to maintain based on my personal activity level.

The 1 or 2 positives to keto for me were improved satiety resulting me thinking less about food and diminished desire/cravings for sweet/salty/fatty concoctions. My calorie allotment is only about 1500 calories as an older, fairly sedentary female. Though I felt more full on keto I had no trouble getting in 1500 calories and had to be careful to not overeat.

For the last many years, my intake is one that most would consider low carb. Certainly not keto. I've not tracked calories consistently for a couple of years but am confident I average around 125 total carbs daily.

To answer your question: I say: don't over think things. Calories are king in regards to weight loss. On the other hand I personally would NOT be inclined to ignore the results of an allergy test without further testing or input from medical professional.

You may like the info in this podcast with Dr. Eric Westman and Amy Berger, two of my go-to sources who have a new book coming out soon. They point out that keto vs low carb really depends on the individual, it depends on your own health, lifestyle, etc. Do what works for you, and you will get results.
https://podcasts.google.com/feed/aHR0cHM6Ly9mYXN0a2V0by5saWJzeW4uY29tL3Jzcw/episode/YjlmY2ZjN2MtNTJlOS00NWYzLTg3ZGEtMmZlZDk4ZTkyYjhk?hl=en-JP&ved=2ahUKEwiytbbCwqntAhVNJaYKHfHrAo4QieUEegQIBhAI&ep=6