The flat battery looks like a LIPO and the round one a conventional Li-Ion. As the others as said it's best not to mix the different chemistries together.
Also the charger inside the powerbank is expecting to charge a 3.7v nominal device so you will need to put two batteries of the same type in parallel (not series) also because you are effectively making 1 cell from 2 or more you need to have cells which are more or less matched in spec so that they charge and discharge in similar ways as you have no way of balancing them individually( you can get these from suppliers already made up!).
Your current power bank should power the RPI for 3-5 hours I think if you need much more then you really need a bigger bank an alternative may be a simple 12V Gel Cell and 5V regulator. I would expect 10 hours from a 5AH cell atleast depending on the regulator type usage etc.
Thanks for sending the pictures BTW as this makes helping you much easier. It's this kind of thing that gets your questions answered in the best way possible .
What is the end purpose of your project?
Good luck !
People are making more of a fuss about connecting different batteries together than is necessary.
For starting a boeing 787 you absolutely need balanced cells of the same chemistry: You're charging them close-to-max, you're discharging them at max when you really need them.
But for hobby purposes where you're adding an extra battery to have a larger capacity, this is less of an issue.
Worst case, what will happen is that one battery will drain from "almost full" to "almost empty" between 3.8 and 3.7V. And that the other will do so between 3.7 and 3.6V. Then during discharge you'll be (mostly) running on just one battery when the voltage is between 3.7 and 3.8, so you'll have to respect the "max discharge" for that battery (and later on in the discharge for the other). Similarly, during charge, you'll have to stick to the "max charge" of the slowest battery.
On the other hand, being extra cautious is a good idea too.
People are making more of a fuss about connecting different batteries together than is necessary.
For starting a boeing 787 you absolutely need balanced cells of the same chemistry: You're charging them close-to-max, you're discharging them at max when you really need them.
But for hobby purposes where you're adding an extra battery to have a larger capacity, this is less of an issue.
Worst case, what will happen is that one battery will drain from "almost full" to "almost empty" between 3.8 and 3.7V. And that the other will do so between 3.7 and 3.6V. Then during discharge you'll be (mostly) running on just one battery when the voltage is between 3.7 and 3.8, so you'll have to respect the "max discharge" for that battery (and later on in the discharge for the other). Similarly, during charge, you'll have to stick to the "max charge" of the slowest battery.
On the other hand, being extra cautious is a good idea too.
Um since my 10Ah power pack was only like £15 and you can get matched cell batteries quite easily I wouldn't even bother taking the risk of mixed chemistries.. Should be OK but since he'll probably be having it unattended then he may as well do it properly!