How can I use a Yongnuo YN-568EX for fill on-camera together with YN-560 III flashes off-camera?

Have you tried mounting the "on camera" flash using a flash bracket? That way it is attached to the camera without using the hot shoe.

Do note that some wireless radio systems have difficulty when a receiver is too close to the transmitter. Some have settings to compensate and allow using flashes closer to the transmitter (at a cost of maximum range)

there has to be a better way!

There is. It's called Godox :) (at least at the time I'm writing this).

Generally speaking, most of the ways to use your YN-568EX with your YN-560 III speedlights will lose TTL/HSS capability on the YN-568EX.

With your existing gear, the main issue is that the YN-568EX has no built-in radio triggering, so you can't use it as a radio master on the hotshoe of the camera instead of using a dedicated transmitter unit. And if you got a YN-685, it only works as a 622 transmitter, not a 560 one. And the YN-560-TX has no hotshoe up top so you can't sandwich it between the flash and the hotshoe. And the YN-560/RF-603 triggering system can't really integrate with any of Yongnuo's TTL (YN-622) triggers or speedlights without losing TTL/HSS function. Yongnuo is slowly moving to an integrated system with their YN-560-TX Pro, but they're miles behind Godox in this regard.

• If you used 603 II/605 transceivers (which do have a hotshoe up top) as your master unit between the flash and the hotshoe, you wouldn't get TTL/HSS with the YN-568EX on top, and you can't remotely power-control the YN-560 IIIs, only fire them (with the 605, you could also turn groups on/off). You cannot use a YN-622 transceiver unit between the YN-568EX, because it can only be a 560/603 receiver, not a transmitter. The Yongnuo 622 and 603/560 systems are mostly incompatible, and Yongnuo's TTL and manual/single-pin gear doesn't really play well together.
• You could get a YN-560 IV/YN-660 speedlight and use that as your on-camera radio transmitter unit, but it won't do TTL/HSS, which is kind of a pain for event shooting.
• Or, you could use a bracket to lift the speedlight up, but you'd have to find a TTL cable that has a passthrough hotshoe up top for the 560-TX. Because using the 560-TX as your radio master, you'd have to add a RF-603 II/605 to the foot of the YN-568EX to trigger it, and it would render the flash manual-only (no TTL/HSS).
• Or, you could set the YN-560 III into S2 "dumb" optical slave mode, and hope that optical triggering from your on-camera speedlight in TTL will trigger them. But. Optical triggering has line-of-sight drawbacks and will be set off by any flash burst the sensor can see, not just the ones from your YN-568EX.

This kind of expansion and manual+TTL gear integration issue is exactly why a lot of us dumped our Yongnuo gear in favor of Godox. All of Godox's gear is in the same triggering system.

To me, the best solution is to sell off your Yongnuo gear and replace it with Godox gear. A Godox TT685 is $110, a V860 II is $180 (li-on battery pack has 3x the battery capacity of a set of 4xAA), and a round-headed li-on V1 is $260. All of them are TTL/HSS capable and can be used as TTL/HSS/remote power radio masters on the hotshoe to Godox's other speedlights, all off which have built-in radio transceivers. No add-on triggers required. The V1 is the only one that comes in Pentax flavor, but all three come in Canon, Nikon, Sony, Fuji, and Olympus/Panasonic flavors.

The Godox TT600 is $65. It's a single-pin manual flash with a built-in radio transceiver, Godox's analog to the Yongnuo YN-560 IV. It can have its power remotely controlled from any Godox transmitter unit (including the TT685, V860 II and V1 speedlights). It cannot be woken up, it cannot be zoomed, but it can do HSS.

And the bigger lights in Godox's system (AD200 and AD300 mini strobes, AD400 and AD600 battery-powered TTL monolights, and the non-TTL AC-powered series II and MS strobes) are all in the same radio triggering system, so expansion options are better than with Yongnuo.