It is possible to take a treaty position if you're a green card holder, but you can’t file a joint return in that case (and you might be opening a can of worms by doing so). You’re required to file the 1040NR each year like a non-resident (that’s why no joint return), with Form 8833 attached to disclose the treaty-based position. But you also have to continue with whatever informational returns (FinCEN 114, 8838, etc.) are applicable, like a resident. (The IRS discusses this here.)
Even more concerning is the fact that since you’re essentially contradicting the idea of the green card with this position (applying the tie-breaker rule of the treaty definitely emphasizes your ties to the other country) there can be possible non-tax implications. david4286608, since we're not able to discuss that here, if you’re still interested in doing this, I highly recommend you speak with some legal/immigration advisors, to be completely comfortable before proceeding!
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