Hey Steve you already know what I think!
If you really want to use the truck, I think you want to first address the brakes. You can make the original brakes work pretty well, but if you are also going to increase the performance you should also upgrade the brakes.
If you plan to move heavy loads and are not real concerned about top speed, the original rear might be okay. If you want to run down the highway with traffic, the stock gears are going to limit cruising speed.
You can improve cruising speed by changing the rear to get better gearing. I have never heard of a gear set available for the original rear that would be an improvement. The other alternative is an overdrive of some sort. A Brownie would be the traditional way to accomplish that, and elsewhere I mentioned a well done small block swap I've seen. That truck does have a Brownie and according to the owner is very good on the highway. I've seen him enough different places I know he runs it all over this area.
If you want to stay with a manual transmission you can use a '55 to '67 (about) V-8 or '64 to '67 six type bell housing AND either modify the stock crossmember and mounts, or replace the crossmember with one from a '54 or '55 1st.
That approach gets your 350 hooked to a stock type 4-speed or a number of other later transmissions and gives you options that might work for you.
The challenge you face is that not a lot of people modify the bigger trucks, so finding someone who has actually done something similar to what you want may be difficult. You may have to figure a lot of things out for yourself.
Someone suggested running a more modern six. A Chevy 261 if you can find one is real close to just a bolt-in swap. Its not a lot bigger than the 216 but it has better oiling and will make a difference.
More work, but looks original and fits the existing bellhousing and transmission is an early GMC motor. A 270 or a 302 will have better oiling, make more power, and in a number ow ways is a superior motor.
More work, and requiring the later bellhousing, is to use the '63 or later 292. This motor is clearly not original to anyone who knows the early motors, but many people will not realize its not original.
Any V-8 swap is going to mean changing the steering. The problem is that the exhaust manifold hits the column. Some guys move it outward to make room, others change the column and box. Another solution is custom headers and there are even some manifold choices that seem to work without trouble. Look for the front dump manifolds from an early 265 for instance.
I hope this all makes sense and maybe even a little of it is helpful. I'm a hot rodder at heart and enjoy seeing others find ways to build the hot rod they want, but as Jeff (RustedNuts) suggested above its your truck and you should do what you want. He is mostly a restorer but still speaks to a committed hot rodder like me.