Your Base to 1-2-3-Tribonacci to Binary back to Your Base
Javascript 117 111 bytes

Thanks to @theonlygusti for helping golf off 5 bytes



x=&gt;{r=0;a=[1,2,3];i=1;while(a[++i]&lt;x)a[i+1]=a[i]+a[i-1]+a[i-2];for(;x;i--)if(x&gt;=a[i]){r+=1&lt;&lt;i;x-=a[i]}return r}


How It Works

First, the function generates all tribonacci numbers until it finds one greater than the input

a=[1,2,3];i=1;for(;a[++i]&lt;x;)a[i+1]=a[i]+a[i-1]+a[i-2];


Next, it reverse searches the list of numbers. If a number is less than the input, it adds 2^(index of that number) to the return value and reduces the input by that number.

for(;x;i--)if(x&gt;=a[i]){r+=1&lt;&lt;i;x-=a[i]}


Finally it returns the result.

Try it Online