Ok, read the guest post on GCC. That guy is still accumulating like me, has 5 kids like me. Makes more than me, but can shelter more, too.
The trick with EITC is that they test income at two points: line 7 wages, and AGI. Your credit is whichever result is lower. So to max EITC, you'd need line 7 wages of at least $13,900, and AGI under $23,750 for MFJ 3+ kids on 2016 tax return. Lower wages drops the credit, as does larger AGI. This would limit any Roth conversion space, below standard deduction plus 5 exemptions.
Your state can sweeten the deal here. My state matches EITC at 30%, and CTC at 33%. When my kids were younger, I've had $11k in refunds, between state and fed, EITC + CTC. I turned around and put that in Roth IRAs for us. Now I've only got one <17 for CTC, and 2 dependents for EITC, plus AOTC. Won't be long before it's just one for EITC and no AOTC.