instrument lighting wiring and fuse
According to this. 4.5 amp max. But I'm no expert.
How to Determine the Fuse/Wire Size
That is probably as good of a quick rule of thumb as any.
The better versions of that are a table or calculator where one also puts in the wire length and how much voltage drop is considered acceptable.
In other words the first consideration is the loss of performance, not the melt dowm of the wire.
One caveat when using tables based on AWG is that automotive wire is not made to AWG. Automotive wire gage is the same outer diameter but made up with larger strands. It has slightly less density and carrying capacity.
My guess as to the weakest connection may be the printed circuit board.
Here's the thing. There were factory options that added lamps to the standard instrument lighting circuit. For example console and tachometer options. On my '67 there was an unused connector for the console option. I connected to it for lighting the tach and oil temperature gages I've added.
Do you know what bulbs are in the circuit? That might be the first thing to check. Sometimes the parts stores have a similar bulb in stock or listed in their books but draws more current.
Instrument lamps probably should be 158. That's rated 2 MSCD and .24 amps
That's the same mini wedge base as 194. But 194 is rated 2 MSCD and .27 amps
Just an example of an easy swap that could pull more noticebley more current when there are 5 or 6 of them.