Just to make things worse for you, the basis of this endevor is to get a little more bass into the works, WITHOUT buying an actual guitar and especially WITHOUT buying a new Blueridge or Gold Tone tenor guitar, as seen on Reverb and at Elderly and supposedly easy to tune to uke fingering.
Because she said something about "TOO MANY INSTRUMENTS ALREADY." Blah blah blah.
I want to stay in that range, but use the more familiar uke chord shapes.
If you OK with just using the same shapes, but having to think for a second what chord you are playing, you can use Chicago Tuning: DGBE.
That is also the standard tuning for baritone ukulele and it's almost the same as standard bluegrass tuning, you just raise the first string from D to E (and no fifth string of course). It's also close to standard plectrum tuning, but then you tune the fourth string up from C to D as well.
You can then play all the same things when playing by yourself, but will have to transpose when playing with others. It makes buying strings easy!
Almost, but I can never wrap my head around the transposition when playing with others.
That's when you slap a capo at the fifth fret.
If you are dead set on playing from GCEA, I would personally start with trying a guitar set, using the fourth string for C, third string for E and the second string for A. I would then use a lighter gauge string for a reentrant high g. Since your plactrum banjo has a slightly longer scale than a standard guitar, it should work fine I would think.
You may very well need to adjust the slots in your bridge and nut.
Disclaimer: I have never tried this, I'm just thinking out loud what I would do in your situation.
EDIT: You could of course also use the fifth string of a guitar set for a low G. Not sure why my spontaneous thinking was going for a high g..
In the 2nd place, it seems as if you could use a guitar 5th(which is "A" at about 24.5 inches, for a nice low G as your 4th, and a guitar 4th as your 3rd, a 3rd as your second, and a 2nd as your first.
Tried Mike's advice, but George is right, that A was all the way to F sharp and half but I knew it wouldn't go any further.
I'll either search for a single or get myself a capo.
What gauge string did you use for the A string? The way I was thinking of it, I would have the pitch of the first string right between the third and the second string of a plectrum or bluegrass set. The second string of a guitar set or the third string of a plectrum/bluegrass banjo set should work fine for this.
I think you might have been trying to get the first string up a full octave higher than what Mike and I recommended? That would put the first string in the same pitch as the first string of a jazz tenor banjo, which would of course be way too much tension.
I've tried just about everything between .009 and .012, everything breaks.
I'm going to suck it up, put on a capo and just be glad I haven't put out an eye.
Tuning to Chicago and using a capo will work fine, but I still think you are missing what I am trying to explain. If you really want to get to GCEA, I would recommend going LOWER in pitch than Chicago tuning, not higher. Try a string around 0.16 for the A.