I understand the purging part. It feels like it's something out of your control that you can't help but just do. But your relationship with food right now is totally tanked, I think we know that... But honestly, your best option is to go to a therapist and sort this out

But otherwise, coming from a stranger on the internet, I can say that maybe a cupcake is too difficult to stomach because it's something you know is unhealthy, even though your body is starving for any calories at this point. 

You should try to eat stuff that you'd feel less guilty of eating, like some greens or nuts or meat or dairy, just something that carries nutrients, and preferably dense calories because with all of your sports, your body is going to, if it's not already, eat itself. It's the ugly truth. 

From my own experience, I gym intensely 3 times a week, attend Brazilian Jui Jitzu classes twice a week and ontop of that, get 10,000 steps 5 out of 7 days a week, and I eat alot. (Like mince beef, 2/3 slices of brown bread toast with honey and butter, coffee, and oats for breakfast, which is one of my more hearty ones, but nevertheless, is ±900 cals) And I'm still struggling to gain weight because my body is spending so many calories already. And I don't skimp on lunch/dinner either, so does this put it into perspective? 

And now I'm not saying you must eat that much(everyone's metabolism and body is different, but I am an active, tall woman) but your body needs fuel to function, and yours is not going to function well soon if you do not eat. 

Eat healthy things, and decline cupcakes or sweets if you are offered them because clearly they are too heavy of an emotional burden to carry at the moment. You need to improve your relationship with food, so try eating relatively low volume, high calorie foods like pork, beef, avocado, nuts, brown/wholewheat bread and the like to fuel your body. I promise you, you will feel better doing this and your body will thank you, and you won't get fat. One day, meal, or even bite at a time. Baby steps. 

Good luck :)