'A proper power source'

Everything is working fine.

My project consists of five devices: A master and four slaves, each based around an RF-Nano.

The RF-Nano in the master is attached to a 20x04 LCD screen and four illuminated tactile momentary switches.
The RF-Nano in the slaves is attached to an illuminated momentary arcade button and a buzzer.

During development, I've used various power sources: A bench supply at around 5v or 9v (which displays a current of up to around 0.05A at 5V, and 0.1A at 9V), or a 9V PP3 battery, connected to the VIN. Or a USB battery back connected to the USB port.

But now it's time to move all this into ABS boxes and make it usable - which needs batteries. I've noticed that PP3 batteries are sometimes frowned upon or considered 'not fit'.

It would be great if I could use rechargeable batteries - but I don't know what sort to get or whether I need some kind of 'control' board between them and my existing set-up.

So I'm wondering: What is the 'proper' power source for this project? Can you help please?

The one that powers it for your required time, and which meets your monetary constraints, neither of which you have stated

Well let's take two example scenarios:

A) Required time is 'as long as possible with a battery roughly the size of a PP3, so that it would fit in the case. Monetary constraint is about 20USD for each of the five batteries.
B) Monetary constraint is 10USD for each of the five batteries, each roughly the size of a PP3. Required time is 'as long as possible'.

The completed project is something a bit like a https://blazepoduk.com/ sort of arrangement. So, it only needs to run for a few minutes at a time, but I don't want to be changing the batteries every half hour or so!

It looks as though a PP3 is rated around 500 mAh. And if I've understood correctly, my devices draw about 10mA. So, if my understanding is correct, PP3s could power my project for around 500/10 = 5 hours. That would be ample.

But I've also seen rechargeable lipo batteries for sale - most commonly in a 3.7V variety. Could I connect two of those in parallel? Do I need a control board for that?

Any help would be gratefully received.

Rechargeable LiPos should be connected in SERIES to increase the voltage. Parallel increases the capacity.

The trick is finding large-capacity 1S LiPos. The largest I own are 600mAh. If you are comfortable with it, you could use LiIons, which are usually around 2500mAh.

I have standardized on CR123A for my low-power projects. Tested them to work down to 5 Fahrenheit (one of my outdoor sensor). If your goal is lowest power and maximum uptime, try not to do any voltage conversion as it is inefficient and waste energy. If you can power the Nano and 2004 LCD directly from 3V, then the CR123A is a good choice.

This one from BestBuy has 2600 mah rating I think