Thinking of building a Tele baritone...

Funny enough, I have been contemplating almost this exact same thing. I have a Tele body that has nowhere else to be, and didn't have a baritone. Only difference was I was looking at an Allparts neck and a B-5 - gotta have a wiggle stick for those spaghetti western riffs. But an Esquire as well.

I fell into a Squier Bass VI and that calmed me down a bit, but if I have some free time this summer it may still happen.

 

I had a Baritele which eventually became the Baritone Jazzlemacaster as seen in my avatar. The neck is a Warmoth conversion, standard slim profile. The Tele body it had is Mighty Mite made of swamp ash. Even though that body is a mite on the heavy side, there was some neck dive with the bari neck attached. That was remedied by moving the strap button from the upper bout to behind the neck, as in the picture below.
I can't help with Bigsby question, having no experience with them at all.

A word to the wise: this is the strap button placement on a left-handed guitar. Make the necessary adjustment for a righy guitar.

 

I should mention that a Jazzmaster-type body balances very well with a bari conversion neck, in case you're at all entertaining anything other than a Tele body.

 

Do it. Now. I went mostly Fender standard OE stuff. For electronics, Fralin T set and CTS, 4 way and a tone bleed. And compensated saddles. A local luthier is doing a bone nut for me, should be ready Monday! First thing I am going to do is ba-dow-dow-down and other trucker song sounds. Who's with me? 27 scale, B-B.
Edit to say the MiM Fender Parts bodies are really heavy. I hung it on a strap built, unstrung, to see if I needed to rethink the pin locations. It balances OK.

 

Oh, for sure a B16 is about the Bigsbiest Bigsby. Like “YO DAWG I HEARD U LIKED BIG ALUMINUM MIDCENTURY MODERN SCULPTURE ON UR GUITAR SO HERE U GO DAWG YO”

 

I've done it and I'm delighted with the results. But there is a twist.

I used a 4-pound swamp ash body, a Fender 27-inch scale length neck (from their Blacktop baritone range) with Fender locking tuners, a Bigsby B5 and a Mastery Bridge.

It's got Lollar Charlie Christians in both the neck and the bridge (called the BS).

I love it. I haven't played it for 10-days but I can guarantee that I can take it out of its case now and it will be in tune.

It's nicely balanced - no neck dive - and is a delight to play.

The twist? It's strung with 12s and tuned E-E same as a 25.5-inch scalelength Tele. I have a baritone Tele (Warmoth 28.625-inch scalelength tuned B-B) so I've no desire or intention to change the Bigsbycaster to baritone tuning.

One piece of feedback and lesson learned. It was previously fitted with a BIG baritone neck (fat back/boat neck proportions) and it didn't work for me as it was a handful with the extra length of the neck. Think shortscale bass with a fat back neck and could you play a 90-minute set with such a big neck.

FYI, my preferred neck for normal scale is a big boat neck.

I hope some of this is useful. Oh, and I've got a B16 Tele which nearly became a BariBigsbycaster.