If it weren't for pointlessly overcomplicated, what would engineers do with all the spare time??
I personally think it'd be quite a bit easier to program certain inexpensive MCUs than an EPROM -- e.g. Arduino or Pololu Orangutan or something BASIC-based... Otherwise... you can implement in all discrete gates but... that is a lot of gates.
Another option... I see that PDIP arithmetic chips are out there so you could explore that. I have never used one so... *shrug* E.g. Texas Instruments SN74F283N at Mouser for $0.72
Basically it adds a pair of 4-bit binary inputs and spits out a binary result with carry.
How you'd use this... ? Couple thoughts. Either one of these chips for each pair (1,2 and 3,4 and 5,6) then take another chip to add the outputs of the first two pairs, and another chip to add the third pair and switch 7. Then a final chip to add the results of these two chips.
Or you could try using discrete gates to add up at least the initial pairs (1,2 ; 3,4 ; 5,6) and the adder chip to go from there. It'd increase chip count but you could say you made your own adder.
Here's a page on the logic/gates involved.
PHY107 Addition using Logic Gates
So use a half adder for each pair. So now you have 3 pair, 3 half adders. Take the 3 half adders plus switch 7 and feed into 2 adder chips (ie, 1,2+3,4 and 5,6 + 7) and feed the results of that into a final adder.
Since you're < decimal 10 (binary 111, hex 0x0A) you can use a BCD (binary coded decimal) to 7-segment driver to drive your LED. Should be able to scare up one of those on mouser easily too.
IIWY I'd design this in Eagle or at least carefully on paper so you can fully utilize all the gates in your quad-ands, quad-xors, etc. And wire them up properly. That is a heck of a lot of chips and pins to get right.
Michael