The reason that diet might work, is because you will create a Carb deficeit which would be filled by Fat, as Fat is a secondary energy source.
However it's not a good idea to remove Carbs, as Carbs help to regulate Glucose levels, fuel the Brain and also the "hydrated Carbons", as they're known, are good suppliers of water, (think Glucose (C6H12O6)), plus the Oxygen helps make better use of the broken down Glucose end product and helps create ATP, an energy source that helps makes your muscles move.
As explained such a diet will not be sustainable in the long term. Not least because most Carbohydrates except Alcohol Carbs are 4 calories per gram, whereas all commonly consumed Fats are 9 calories per gram, so the risk there is, if you actually added back the same grams of Fat as you took out with Carbs, you'd consume a lot more calories. More than twice the amount on a Carb to Fat comparison.
4:1, Fat:Protein doesn't sound too logical.
If you had just a sedentary females daily requirement for calories, (1,800), men are 2,000 a day, then you'd be eating 360 calories of Protein a day, (90 grams), and 1,440 a day from Fat, (160 grams). It's reccomended as an RDA, you consume 70 grams of Fat on a 2,000 a day diet, with 20g being a limit for Saturates. So you can see why this diet of yours is out of whack. You're eating more than double the amount of Fat a sedentary man should get.
You're also exercising, so aiming for a sedentary females caloric average would be a bit hit and miss, when you could be having more than that, to give you calories to burn through exercise and encourage better weight loss.
Comsuming a lot of butter doesn't make entire sense. Butter does contain Lauric Acid, which is present in Coconut Oil and is a "medium chain Fatty acid", that ends up in the Liver and converts to energy, so it doesn't store, but Coconut Oil also contains Capric and Capryllic Acid neither of which I believe are in Butter, but I'm not 100% certain of that.
Plus havign a slew of Saturates isn't necessarily the answer.
Olive oil is good though particularly for Alpha Linolenic Acid, (Omega 3) and being Omega 3, it partly breaks down to EPA and DHA the Fats in Fish, which is beneficial, as Fish is a carrier of Heavy Metals, which can become toxic at high levels and cause Nervous system issues.
Cold Pressed Extra Virgin Olive Oil is reckoned to be the best.
As suggested, bin the diet immediately.
Cassie mentioned more than enough for you to mull over, so I'll say no more.
Good luck though with everything and we will be here if you need anything else
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