I have a bass instrument track that has a bass note every eighth-note.  
 
I also have a synth track with some long notes that I want to create a "pumping" effect at quarter-note intervals.
 
What's the best way to do this? 
 
I already tried a few things but didn't have success:
1) Tried putting a sonitus compressor on the synth track and sidechaining in the bass track, but that resulted in pumping every eighth-note.
2) Tried automating the volume with a sine wave, but I find that tool very hard to create a curve with consistent amplitude across the entire track.  Like I couldn't figure out vertically where to start drawing the sine wave so that the amplitude modulates the right amount and at the right level consistently.
 
I know I can copy the bass track and create a new instance of the synth, change it to quarter notes and then use that as a sidechain input for the synth, but that solution isn't very efficient. 

You could create a track with anything playing quarter notes (kick for example), and mute it.  Assign a pre-fader send to the compressor side-chain.  Similar to what you did with sending from the bass track, just from something that only hits quarter notes.
 
This is probably an overly-complex way of getting quarter-pumps, but it's the first that comes to mind and should work.
 
 
Edit: I only read your last line after writing this.  Great minds...  

The easiest way I've found to do that is to use a tempo-synced tremolo plugin. Meldaproduction has a free one that works great and lets you modify the modulation waveform so you can make the effect as subtle or abrupt as you like.

bitflipper
The easiest way I've found to do that is to use a tempo-synced tremolo plugin. Meldaproduction has a free one that works great and lets you modify the modulation waveform so you can make the effect as subtle or abrupt as you like.

That's a good idea.  I'm not a huge fan of Melda because the interface is a little bit complex for my tastes.  I might try to find a standalone free tremolo plugin rather than downloading and installing the Melda free bundle. 
 
Regarding the sine wave tool. . . is it just me or is it really hard to use in this context?