The way I deal with the taper at the nock end is to use 5/16" or even 9/32" nocks. (Next time you're at a Trad Shoot and a vendor's there selling nocks, by a dozen of every size.)
Also, to strengthen the thinner nock end of the cane I will drill out the first node closest to the nock end with a 5/32" drill bit. Then I'll take one of those large bamboo shiskabob skewers (the 8" ones, not the 6" - you can buy a pack of a hundred for 99 cents at your local discount store), lube it up with wood glue, and insert it into the cane point-first all the way in (or as far as it will go until it hits the next nock).
Even though they're thin, I have NEVER had a cane arrow break on me, even when stump-shooting steel drums! I can't say that about any other arrow material, including carbon...