The only difference in a 2.75" shell and a 3" or 3.5" in really the number of pellets. The range is the same. Many turkeys have been killed over the years with 2.75" shells.
A good swarm of #6 shot or #7 shot will certainly do the job for you at ranges up to 40 yards. The #5s are limiting the number of hits and also can produce unacceptable gaps in your pattern because of that.
I'd recommend that you consider another turkey choke for your gun. That could make all the difference in the world in the quality of your patterns. A good dense 10" core pattern, without large gaps and a decent "halo" pattern around that, out to maybe 16-18" will do the job every time.
There are many good chokes out there, if you are a member of the NWTF, you may want to take a look at the March/April issue of "Turkey Country" magazine and look at the article I wrote "Choking Winchester's Long Beard XR" and take a look at the chart to see what chokes worked best in with the Invector+ barrelled guns like yours. I did use 3" shells but the principle is the same.
Please let us know how it goes and all my best in the season.
Thanks,
Clark