If you're drawing 27" and using a 28" arrow, the 340's would be pretty stiff. While I enjoy making, shooting and hunting with wood arrows, I think the Easton Axis arrow is one of the finest on the market. I use a .400 Easton Axis shaft from a 59# @ 28 inch Super Shrew longbow that's cut 1/8 inch from center. The arrows had the 16 grain insert and 200 grain broadheads and arrows are 29.5 inches to back of point (BOP). Total arrow was about 550 grains and is like a runaway freight train. You're shooting less weight and if drawing 1 inch less, this will require less spine. It appears as though you're shooting a recurve and if its cut to center or past center, then that will require a stiffer arrow. My hunch is that you will need 400 size at most, again depending on lot's of variables (centershot of bow, how long you leave your shafting, ect.). You should not be paying extra for inserts as they should come with the shafting-unless you're upgrading to a brass insert for more weight up front. Most places will charge for cutting and installing inserts however. I get all my Axis shafting from Kustom King arrows (sponsor here). Tim Cosgrove, owner, has used Axis shafts for years now and is very knowledgable about them. Also, Bill Carlsen here on Tradgang is another Axis user who has a lot of experience, so listen to his advice if he posts. I've been using them out of my bows for six or seven years now and so I've had some experience with them too.
I'm currently using Axis 600's with 250 grains up front out of a 44#@28 inch Super Shrew with 29" BOP arrows. They fly perfect and if you look at some of the charts, they would call for a 500 size Axis shaft. It would be good if you could get ahold of a 400 shaft and experiment with it first or perhaps inquire on getting some used on the classifieds here to try before investing in new.