my opinion!!

I think it is not a big deal since EIC is not a welfare. You are entiltled to get it based on tax law.
If you follow tax law, you will be fine. No need to give up your legal right because of Green Card issue.

But when you apply for your green card, make sure your income for your whole family must well exceed 125% of poverty line set for the state where you live.

Just for reference!!