Can I claim one child for child tax credit, but put both children down for EIC if I have full physical custody and their dad claims the other for child tax credit only?

If you are the custodial parent you can still get EIC and you can also (if this applies to your situation) get the childcare credit, and Head of Household filing status.   The non-custodial parent can get the child tax credit for the child(ren) under the age of 17.   Non-custodial parent does not get EIC for the child and non-custodial parent does not get childcare credit or HOH filing status for claiming the child.

Are you the custodial parent?  Do you have an agreement with the other parent to allow the other parent to claim them--due to divorce or that you live apart and share custody?  Did one of you sign a Form 8332?

If there is a signed 8332 then the custodial parent retains the right to file as Head of Household, get earned income credit and the childcare credit.  The non-custodial parent gets the child tax credit for children under the age of 17.  If the child is 17 or older the non-custodial parent gets the $500 credit for other dependents.

If you and the other parent have a signed agreement, you need to indicate in MY INFO that you have such an agreement.

As far as the IRS is concerned, the custodial parent is the one with whom the child spent the most nights during the tax year--at least 183 nights.