Is there processing gain for FMCW using heterodyne-style receiver as opposed to matched filter?
Beat signal of a single target will be a sinusoid in the idealized world,

Only for a TX waveform that has a constant slope in frequency domain (that means only with sawtooth-shaped frequency trajectory).

so theoretically the signal processing gain of an FMCW pulse correlated with Tx waveform in this way should be analogous to FFT processing gain of a single tone, right?

No, because the FFT gain is below the theoretically maximum processing gain, in general (it is the optimal estimator only for beat frequencies that are multiples of (sample rate)/(FFT length))
The FFT is usually not a good frequency estimator if you're just looking for a single tone.

Just trying to wrap my head around what to expect in the field when playing with waveform parameters and what kind of SNR before/after I can expect to see. Is it really just a product of the duration of wave form and bandwidth?

Yeah, the integration gain basically collects all received energy.