Betta Advice Needed. Please Help.

Welcome to the forum.

A heater and filtration are an absolute must. Rather than type lots of things out, here's how to look after Bettas :General Betta Care

Needed Tank Equipment is probably the section you need, but it's all a good read for Betta care.

Hope this helps.

Some fish shops really annoy me with that 'they only live in puddles' twaddle.

I'm not getting at you (we are all guilty of impulse buys though research before buying in future  ) but this is utter rubbish that some shops spout. Bettas live in paddy field-like conditions in the wild. These are vast areas of shallow, slow moving water. They dry out in the dry season, and a few bettas survive in the left over puddles until it rains again. Next time, ask the shop if they live in puddles, how do the males and females ever meet  Sorry, rant over.

Bettas are tropical fish; they need a tamp of 25 to 26^oC, so in the UK they do need a heater unless you keep the room at that temp 24/7.
They can be kept in unfiltered tanks, but they are better with a filter and it's easier on the fishkeeper too. In an unfiltered tank they need 50% daily water changes with a 100% change once a week. Yes, they can breathe air, but they are still poisoned by ammonia and nitrite like other fish.

As for where to keep him, he could be kept in the 23 litre tank, but in your case I would advise against it. A lot of bettas don't get on with guppies as they mistake the long flowing tails for another betta and attack. And although the cardinals aren't usually nippy fish, in a small tank there is always the risk they will nip and the betta would have nowhere to escape.

Is there any chance of getting another tank? To be honest, something like a 40 litre tank? And move the guppies & cardinals into that and put the betta in the 23 litre. I suggest this as cardinals are better in a tank bigger than 23 litres. At the least, another small tank? I have a 25 litre tank for a betta (I'm currently doing a fishless cycle in it, long story!) and I like that size tank for them. They need things to swim though and a cave or two. Plants should be real or silk, not plastic, and other decor needs to be smooth with no rough edges. Plastic plants and rough decor can tear a betta's fins.
You can do a fish-in cycle with a betta more easily than with other fish simply because they can breathe air. But even so, enough water changes need to be done to stop ammonia or nitrite ever getting above 0.25ppm.