I'd try this for the transmitting coil (if you have some litz wire you could make your own, but the ferrite film behind it helps a bit)
A fine starting point. L may be on the low side, but the general idea is right if nothing else -- that's just a matter of turns.
and this as the receiver (really high inductance for the size, very high permeability ferrite). Make the transmit coil resonant with a series capacitor and connect it to a square wave generator of variable amplitude, put the smd inductor in parallel with a CG0 capacitor and a LED and see what it all does. The chip inductor probably has to point towards the transmitting coil BTW.
Now, the drawings don't show which way this part is wound, so it's not clear how good it would be. It's "unshielded", apparently, but beyond that...
If it's wound axially, then that gives L/D ~ 2, and that also means the end terminations are shorting out the field at the ends. Which both shields it from external fields, and reduces inductance and Q significantly.
They don't even rate it for Q, only DCR. ACR is typically several times DCR (2-10 times, say). Probably since thin wire will be used here, core loss (and end cap electrode resistance) will dominate, which can also be expressed as ESR. So, it's probably not all that great, electrically.
It's also not clear how high permeability is. This is a useful ref:
https://www.seventransistorlabs.com/Images/Rod_Core_Pressman_Billings_Morey.pngFor L/D ~ 2, mu_eff doesn't increase much for mu_material > 10: it could be low-mu powdered iron, or high-mu ferrite, and you wouldn't know the difference. (Hopefully they'd use a lower mu material, so that core loss isn't terrible?)
Tim