Thanks Curt! I'll continue to make updates until this thing is finished and belting out some tunes.
Newest update is the veneer is now on the lid, and here's how I did it:
First I did some research on how do this patchwork veneering, and liked some of the methods this guy has in his video:
https://www.youtube.com/watch?v=r0QDfifQgkk
If you didn't watch it, he used a piece of flat aluminum pipe with sandpaper glued to one side, and sandwiched the two matching pieces of veneer between some flat boards so that he could sand the edges to get a perfect fit when glued down. I didn't have an aluminum pipe, but I do have a jointer and some pine wood so I put some 120g sandpaper on that with some spray glue:

- Sandpaper glued on one side of sanding block
Then tapped the matching veneer pieces back-to-back:
Then sandwiched it all between a couple boards (one of which is oak plywood) with just a smidgen of the veneer sticking out:

- ready to sand (or plane)
And used my new sanding block to even out the edges of veneer, just like the guy in the video:
The results of that process were just as I suspected:
With the edge matching done, I then used blue 3M painters tape to join all the pieces together (again, like the guy in the video). This is done on the good side that faces up since I'll be putting glue on the back:

- Pieces tapped together
After everything was cleaned up, I started to heat up the hide glue with my heat gun to make it good and runny:

- Heating up Hide Glue
I dumped glue all over the lid and took a bit of time to smooth out all the glue with a little spatula. I was concerned the glue cooled down too much so I reheated it with the heat gun for a few seconds:

- Re-heating the glue
I then tossed the Frankenstein veneer on there and used this little roller to stick it down. I know there's a veneer hammer tool for just this purpose, but I didn't haven't one and was worried that pulling on the hammer would just tear all the pieces apart. The rolling is just a temporary fix before the fun part:

- rolling it out
The fun part is the clamps. I have two ½" sheets of MDF that I'm using as sandwich boards to squeeze everything together. Used all my clamps again:

- Clamps!