Your headline asks about the child tax credit.
Other income such as the PFD does not change your ability to claim your child as a dependent, as long as the child's income is not so high that they pay more than half their own living expenses. (And even if a child makes so much money that they could pay more than half their expenses (a child TV actor, for example) it still doesn't count as support unless they actually use it for living expenses; if they save or invest the money, it's not support and they are still their parent's dependent.) The $2000 child tax credit is just based on having a child dependent under age 17. If you were to file MFS, one parent could claim the child and get the $2000 child tax credit.
Your main question also asks about EITC
You are ineligible for EITC if you have more than $10,300 of investment income. This is substantially increased from the past. If you file MFJ to claim EITC, your $6600 PFD income will not disqualify you from getting EIC, as long as you have less than $3400 of other investment income.
If you report the child's PFD on your tax return, you would have $9900 of investment income, which is still less than the $10,300 threshold, so I think you are fine. (I don't actually know if the child's income counts against you for EIC in that situation, but even if it did, you are under the newly increased limit.). There's no harm in reporting the child's PFD on their own return, though.
