Gain muscle! You won't become 'bulky' - you'll just feel fit. Plus you'll be able to eat 3000 cals per day.
Unfortunately this would only happen if you gained enough that you could outlift The Mountain on game of thrones.
That's not true at all. Gaining muscle means your RMR goes up. Which means if you have muscle (not bodybuilding standards by any means), you can train your body to take lots of lovely calories and burn them off doing nothing at all. You won't get ripped doing resistance training unless you have 0 fat in your diet.
Instead of eating dust and running yourself into an early grave to be skinny, you can eat more and more calories, resistance train and be healthy, slender and toned.
I don't understand the desire to be skinny at all.
And BMI is a terrible tool to measure your size without other factors taken into consideration.
So in answer to the OP, it's perfectly reasonable to have and want both. You don't need to starve to be thinner and certainly not healthier. But to achieve such you need a lifestyle change, not a diet or excessive cardio which wastes away your muscles.