My goal is to get a smooth sustained DC signal at 1.4V if I have a 50% duty cycle.
This is my circuit so far, where R5 & R7 reduce the PWM signal so it lies between the op amp's 0.3V - 2.5V rails.

The LM358 is not a rail-rail type op amp.
It has a maximum output of about 1.5V below the supply rail, so if you want 2.8V maximum output. the supply voltage should be 5V.

Below is the LTspice simulation a simple RCRC LP filter circuit that should do what you want:
You don't need an op amp since the Vset input has a high impedance.
From the data sheet:
"Leave floating, internal current source (typical 1.4µA) will pull up this pin to 4.5V"
This implies an input impedance of at least 3.2MΩ.

The output is 1.42Vdc for a 12V, 1kHz, 50% duty-cycle input, with a rise-time of about 22ms.
Of course the circuit will work well for any PWM frequency higher than that.

 

It doesn't utilize the full PWM scale though (0% = 0.3V; 50% = 1.4V; 100% = 2.5V).

How did you arrive at those voltages?
To get those values you would have to add some offset and non-linear gain.
PWM is linear so, for example, if 50% duty-cycle is 1.4V, then 0% would be 0V and 100% would be 2.8V.

Mind if I ask where you got your experience in filter design?

Mostly from reading about filters and simulating various designs in LTspice.
Spice is a great analog design tool for experimenting with various analog designs, including learning about the response of various filter types.

 

It is just 0.3V to 2.5V, so 1.1 is it's 50%. So I think I should be able to just play around with component values to modify the filter.

It requires more than just a change in component values, since you want a non-linear output from a linear input.
If you really need those points to correspond to 0%, 50%, and 100% PWM duty-cycle, then you will need to add an op amp at the filter output with some offset and non-linear gain.

Do you really need the output to exactly correspond to those input duty-cycles?
Otherwise just make 0.3V=12% duty-cycle, 1.25V=50% duty-cycle, and 2.5V=100% duty-cycle.
Or with 0.3V offset, 0.3V=0% duty-cycle, 1.4V=50% duty-cycle, and 2.5V=100% duty-cycle.