First, If you are concerned about reCAPTCHA presenting a roadblock to conversion take a look at reCAPTCH v3 before you do anything else. The new hotness: reCAPTCHA v3 | reCAPTCHA | Google Developers
Otherwise...
For reCAPTCHA v2 there are three callback functions that can be specified via html custom attributes on the widget.
- One gets called upon success from the service.
- Another function for the expiration of a successful verification.
- Another for error handling.

You could specify the callbacks as custom functions that you write, or just call straight into _satellite.track("foo").
It looks like some implementations just use a DIV with a form submit which you should be able to catch as a DTM custom event if you wanted one more metric beyond those provided by the callbacks.
And.. just to be complete, if you are doing explicit rendering of the reCAPTCHA widget, you could capture the render event by calling _satellite.track within that function.
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A parting thought. The scenario that you propose is perfect for A/B testing if you have the traffic volume to support it. Testing different reCAPTCHA treatments head-to-head would give you much clearer data than fallout analysis of different treatments in different timeframes.
Cheers,
-Stew