Fuse blowing during capacitor charging can be prevented by using a "slow blow" fuse which tolerates moest overcurrent peaks of say 5x rated value for a second or so but which blows much more rapidly under heavier overloads. Getting the right balance between allowable overload and good protection can be an interesting exercise.
Slow Blow Fuses:
A slow blow fuse will take seconds to minutes to "blow" at 2 X its rated current but will typically blow in 10's of mS at 10x rated current and in under 1 mS at 100 x rated current. Exact current time profile will vary with brand and type. This time delay effect is usually but not always achieved by giving the fuse "thermal inertia" and by controlling the cooling available to the fusible element.
The graph below shows the current versus time profile for Littefuse HF461 fuses.
This is NOT the cheapest slow-blow fuse available. They can be purchased for rather less that the $1.50/10 or so per fuse that these cost, but these have a good data sheet to show you what you can expect. (Or hope to expect).
Below it can be seen that the 0.5A version will blow in about
20 seconds at 1A (2 X load)
0.6 s at 2A (4X overload)
50 ms at 5A (10X overload)
Under 10 mS at 10A
Under 1 mS at 30A
In the case in question a 2A fuse with a 6A load should blow in 1 to 2 seconds.
These results will depend to some extent on temperature, mounting, enclosure and which way the wind is blowing (in some cases literally so) but will serve as a guide which should be good enough..
Slow Blow fuses.jpg
Start your Digikey search here
Click on SlowBlow and then set filters
Here is a $US0.20/10 offering
bel 5ST / 5 STP series
Also provide a good data table:
Note the very long periods when Iin is not much > Irated.
![enter image description here](https://i.stack.imgur.com/xfTtW.jpg)
I would perhaps use a relay, with discrete components you won't necessarily get good accuracy. If you don't want a huge relay hanging around, I would use a comparator with voltage reference. I'd say using discretes in this case won't really cut it, unless you aren't looking for accuracy. Another option is to build the comparators out of transistors, but that sounds pretty painful.
Best Answer
I have had several students who have used super cap chargers. They provide a lot of nice features like variable current limit on the charging, the particular one that I linked to will also make it so you could use 2 caps with lower voltage on each one and it will make sure you don't over voltage one.
The only down side is the chips for these are VERY small and can be difficult to solder if you aren't very experienced.