Well I am not sure why you can not have two antennas, one for 10 and one for 12 meters because I do.
I have an HF-Auto tuner in line.
Antenna port one has the coax for my 80/40 fan dipole connected to it.
Antenna port two has my Hy-Gain TH2MK3 20-10 M yagi connected to it.
Antenna port 3 has a remote coax switch connected to it which handles a 30 M dipole, 17 M raised vertical, 12 M delta loop and 6 meter yagi.
So if I want to operate on 12 M I use HF-AUTO port #2, put the remote coax switch on the relay that the 12 M loop is connected to on the remote switch mounted on the tower.
Key amp, amp goes to 12-10 meter settings and I am good to go.
My 3 HF rigs are all connected to a 4 position coax switch. From there the coax jumper goes to the amp input. Then from the amp Antenna 1 goes to the tuner.? I use Antenna 2 on the amp for my dummy load for testing purposes.
All of my HF rigs can key the amp by using a DX Engineering Dual Keyline.
So I am not sure why you can't have two separate antennas for 10 and 12 meters.