Building a charger for a capacitor
A current mirror is useless for this purpose.

Have a look at Julian Ilett's channel on YouTube

He likes to do stuff with supercapacitors and has several videos on the subject.

In general he uses an adjustable power supply module with adjustable current limit. He uses a powerful one which will be overkill for your problem.

I would use a module like this. Which can convert 5 V into 2.5 V and where the maximum current can also be set.



Since this is a switching regulator you can even charge with a higher current at the output, up to 2.5 A or so, without overloading the 5 V 1.5 A supply.