Ahoy and welcome.
We're currently liking the SkyRC S65C:
https://www.nexusmodels.co.uk/skyrc-s65-charger-ac-balance-charger-discharger-65w.html or elsewhere.
Batteries, I'd go with 7.4V lipo, although lithium-ion owners seem happy enough too. I'd avoid 11.1v as while you'll get better trigger response and ROF, you also get trigger arcing and may get overspin and double-firing on semi (some people get lucky with 11.1v, I've not been).
It looks like the SA-C05 CORE uses mini-Tamiya (yuk) rather than Deans. Mini-Tamiya is the left here, Deans is the right.
If you do have mini-Tamiya, you be sure that you're buying
airsoft rather than
RC batteries, as the connector polarities are reversed for reasons known only to God (Deans doesn't have that problem, which is another reason to rewire to it).
You'll want a battery with two distinct cells, which you'll see called a cranestock / nunchuck / butterfly style. One goes down either half of the stock after you've removed the back-plate. I'd measure the space that you have available and buy two of the highest capacity lipo batteries that fit, within your budget. For the same price you'd be able to buy
one lithium-ion battery with about the same capacity as two lipos, but I consider that a riskier proposition if it craps out. You'll generally want to swap lipos at lunch, unless you're light on the trigger, but you can eke out a day on one.
For example:
https://www.componentshop.co.uk/7-4v-1100mah-20c-continuous-discharge-airsoft-split-pack-lipo-battery.html
https://www.componentshop.co.uk/7-4v-1300mah-15c-30c-lipo-airsoft-battery-2-way-split-giant-power.html
https://www.componentshop.co.uk/7-4v-2200mah-20c-continuous-discharge-airsoft-split-pack-lipo-battery.html
I'm
not guaranteeing that all of any of those will fit down your stock, you'll want to stick something down each side and measure it to be sure.
https://hobbyking.com/ can also provide batteries, as can eBay or Amazon, but I'd hesitate to recommend any specific seller as with stocking levels the way they are, you might get stung with surprise import duties if they ship from abroad. I've not bought any for a while, so I can't give a firm recommendation as to brand or retailer.