There is no dietary requirement to eat carbohydrates.
Your body and muscles can run quite happily on fat. That's what fat is for.
It is true that your blood needs to maintain a minimum concentration of glucose as your brain cannot run on fat. However, that does NOT mean you need to eat carbohydrates.
You know how insulin converts glucose into fat? The process can of course be reversed. Your body can convert both fat and protein into glucose (without insulin) so provided you eat enough fat and protein, and you've adjusted your insulin intake to account for your carbohydrate intake and basal metabolism....you can quite happily eat 0g of carbs a day and theoretically encounter no problems.
There is of course a big 'if' over all of this as manually matching your insulin for this can be challenging (as you've already found with your basal), and it's worth pointing out that just because you can go without carbs for the entire day doesn't mean you must. But I would say that one of the more successful approaches to managing T1 is based on a maximum intake of 24g of carbs a day with no more than 12g in one meal and there are people in their 80s who've been following that regime for decades.
So, as a T1, it is really up to you how many carbs you want to eat. You can have 0, you can have 500, you can have anywhere in between - but the more carbs you add, the more action you need to take and the less control you will have. Again, to rehash the driving analogy, more carbs = driving faster, with all the risks that entails.
