problems arise when Bank 1 has memory

You didn't explicitly say so, but I'm assuming that you swapped the RAM around in order to narrow in on the slot rather than a bad memory module?

If so, problem could be with slot itself or elsewhere on the board. In either case, Apple's approach would be to replace the entire logic board.

Maybe check around your city for a shop that does board level diagnostics and repair? You'd probably be looking at a few hundred dollars doing it that way.

Honestly, I'd just stick a 4GB module in the good slot and call it a day.

 

As noted above, to 'complete the trouble-shooting analysis' you would have to confirm whether you DID Swap 'Slot 0 and Slot 1' RAM, with same '3 Beeps' being the result.. ** That tone burst is the indication of RAM/ Slot Error..

I know this only because my 2011 MBP had this exact issue ~ No matter what RAM stick I would insert into Slot 1, it would 'error', but could limp along on Slot 2 just fine!! I did for a week until my local Apple Repair Depot got parts in (pretty sure it was a Logic Board Replace, Applecare was my friend) and got it back a day later, has been great since.. I recall that the piece was 'listed' as ~$900 item though.. OUCH indeed!!

 

So you bought new RAM and it is working again?

I'm asking because I have the same problem, after some trial and error I figured out that having RAM in the lower bank will lead to three beeps eventually, while the top bank seems to be working fine. Now I'm not sure whether to try and go for new 2x4 GB or whether I should just buy 1x8 GB to get back go my 8 GB config.

 

If could be dust related.

Disconnect battery, blow slot 1 out with compressed air, and with a fine blade spudger wrapped in a lint free cloth with a few drops of denatured isopropyl both sides clean the contacts out and blow with air again. Let dry for 30 mins - 1 hour then power it up see if slot 1 works. If not it's logic board unless you can re-ball SO-DIMM slots!