Yes. That's why I chose this chip
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I guess my question wasn't clear. I'm asking if I can have the MCP73812 *and* my load hooked up to the battery at the same time. The diagram in the datasheet (which you posted) shows only the MCP73812 connected to the battery and doesn't discuss its interaction with a parallel load which is also powered by the battery.
I presented the question with a generic "charger IC" because I felt like it shouldn't matter what the charger IC is since all of them I have seen are meant to be used in the manner that was shown in the MCP73812 datasheet which you posted.
Actual Question: I'm wondering if I need to implement some sort of switch to disconnect the load when the battery is charging (like my diode auctioneering circuit in the first post) or if I can just leave it connected and the charger will be ok (like the first picture in my first post).
I'm also unsure if perhaps power is meant to flow backwards from VBat to VCC to power the load (so the load goes on VCC, not VBat), but since there appear to be protections against this since the MCP73812 shuts down when VCC < VBat, I don't think that will happen.
Part count is important with my project and less parts means less headache laying it out. I'd also prefer to avoid the diodes because that may push the input voltage to my circuit too low.
Most "examples" on the internet are for RC vehicles and thus don't need to worry about this because the batteries are physically removed for charging. That's why I'm asking this question: because I can't find any examples of keeping the load (which was powered by the battery) running while the battery is charging.