I was asking for the slope of oscillator tuning in Hz/V, not for the DAC voltage range. I mean you want to know what size of deviation you are looking for. The tuning voltage deviation may be 1 mV, 10 mV or more.
The capacitor on the reference voltage is shown in the DAC data sheet. It is missing in your circuit (design error).
Anyway, most engineers use a PWM type DAC for this application, for better differential linearity. Equally small frequency adjustment steps can make a difference. A MCU with 20 MHz can provide a near perfect 24 bit resolution in one second.
Those schottky diodes are another design error. I would replace them by a small resistor, like 39 Ohms. And route the current from R1 through that resistor, too. This way you can tune the output +5VA to a stable 5.3 V.
You possibly have Gnd from antenna, supply (PE), counter and scope/computer. Sometimes one can see ground currents with the scope, measuring some mV between different Gnd points.
Regards, Dieter