No, that will not help you use the credit more quickly than you expect. The 5695 is a non-refundable credit. That just means that you cannot reduce your tax liability below zero with it. Since you have no tax liability after the credits are taken, you could reduce your withholdings/estimated taxes, but that won't change the credit carryover.
You can take the credit up to your tax (do not include estimated payments). As a an example with fictitious numbers, say your tax from Form 1040 line 24 was $2,100 and your Child Tax Credit was $2,000. You could take $100 from your 5695 and carryover the rest. Whether your employer withheld $100 or $10,000 for federal income tax, you would still get the total withholding back. The credits wouldn't change based on the payments.