You could also do the 8 minute protocol and take 90% of that for your FTP. The way that one works is that you do two 8 minute max efforts, with plenty of rest in between. You don't average the results, you just use the better of the two.
Hermes makes a good point too though. 20 minutes is totally arbitrary, and you could just use 12 minutes. Use 90 or 92% and see how it goes.
(I had more thoughts here)
If it was me, I'd use that 13 minute climb as your test. Forget about 20 minute or whatever. As Hermes said, start with 92% and make that your "FTP". Then whenever you do an FTP test use that same hill and stick to your 92, 90, or 95 as the percentage for setting your number.