Convert to Bibi-binary
Python 2, 58 bytes
f=lambda n:(n>15and f(n/16)or"")+"HBKD"[n/4%4]+"OAEI"[n%4]
A recursive solution. Try it on Ideone.
05AB1E, 20 18 16 bytes
hv…ÂkdžM¨ÁâyHèJ
Explanation
h # convert input to hex
v # for each
…Âkd # string of the possible first Bibi-binary letters
žM¨Á # string of the possible second Bibi-binary letters
â # cartesian product to produce list of Bibi-binary pairs
yH # convert hex char to base 10
è # use this to index into the list
J # join
Try it online!
Saved 2 bytes thanks to Adnan
Python 2, 81 76 bytes
lambda n:''.join('HBKD'[int(x,16)/4]+'OAEI'[int(x,16)%4]for x in hex(n)[2:])
Chooses the bibi-digit to represent each hex digit based on the patterns in the bibi-digits.