One YN622C on my camera.
One YN622c on my YN656EXIII.
Two YN600EXRTII triggered by the YN622C on my camera.
Would I be able to do that...?
No. The Yongnuo 622 2.4 GHz triggering system is incompatible with the Canon RT system Yongnuo's cloned for their RT gear. To use a YN-622C or YN-622-TX as your on-camera transmitter, the YN-600EX-RT flashes would require YN-622C transceivers to be attached to their feet. Their built-in transceivers are only good for the Canon RT system.
or would I need to get a different radio, like a YN-E3-RT II?
Yes, to use the two RT speedlights, you would need an RT-compatible transmitter, but it could be that non-Yongnuo or non-Canon units (like, say, a Jinbei TR-Q7 or a Phottix Laso transmitter) would not work. 3rd party+3rd party can often have additional incompatibilities not seen with same-brand or OEM gear.
If so, how could I fit my the YN565EXIII within that setup?
By adding an RT-compatible receiver to its foot: say, a YN-E3-RX.
However. What I would recommend (at least in 2021) is that you drop the idea of getting Yongnuo YN600EX-RT flashes. I would recommend that you ditch your 568EX+622 setup altogether, and go with Godox flashes instead.
At this time, Godox makes a $110 speedlight, the TT685-C. It's a reasonable near-clone to a 600EX-RT. It has a built-in 2.4 GHz radio transceiver, it does eTTL II and HSS with a Canon body. But that 2.4 GHz transceiver is in Godox's radio triggering system. Which, unlike Yongnuo's does the following things:
It integrates with manual speedlights. The Yongnuo 622 and RF-603/605 YN-560-TX systems aren't fully-integrated in the same way. To use 622 gear in the 603 triggering system, you lose TTL/HSS capability. With Godox? A $60 TT600 works just fine without making you ditch TTL with a TT685 in exactly the same way you can't work a YN-560IV in with Yongnuo's 622 or RT gear. And, unlike a YN-560 IV or YN-660, a TT600 can do HSS as a radio slave.
The TTL/HSS support in Godox's system works (with some minor exceptions) across brands. And Godox supports Canon, Nikon, Sony, Fuji, Olympus/Panasonic and Pentax. Yongnuo only supports Canon with the RT system, and Canon and Nikon with the 622 system. And that 622 support? does not work across the brands.
Godox has much bigger lights than speedlights in their system, scaling all the way from mini speedlights to 1200 Ws pack'n'head strobes. Expansion into proper studio lighting is easily done, and integrates well with your speedlights. There's a lot more room to grow in this system. Yongnuo only makes speedlights and a non-modular Godox AD200 clone.
While RT gear does have some function that the Godox system doesn't offer (remote wake-up, 4-digit ID codes, a channel group, and remote shutter function via the flash hotshoe), theses are gracenote features that are typically worth trading off for a $60 HSS radio slave flash, and the option of bigger lights. Godox, IOW, is more of a poor man's Profoto than Yongnuo is.