sorry but the tax rules say that under the conditions you presented only one of you is legally entitled to claim both kids and that's the one with the highest adjusted gross income. This has a substantial economic effect because the one that can claim them gets HOH status while the one that can claim no dependents must file as single.
the way you want to do it is that both of you would be able to file as HOH with a qualifying dependent but that's not according to the tax laws and if caught there would likely be substantial additional taxes, penalties, and interest. The wrong person claiming a dependent and thus getting tax credits they're not entitled to could be barred from claiming certain credits in the future that can range from 2 to 10 years.
IRC 152(c)(4)
(4)Special rule relating to 2 or more who can claim the same qualifying child
(A)In general
Except as provided in subparagraphs (B) and (C), if (but for this paragraph) an individual may be claimed as a qualifying child by 2 or more taxpayers for a taxable year beginning in the same calendar year, such individual shall be treated as the qualifying child of the taxpayer who is—
(i)a parent of the individual, or
(ii)if clause (i) does not apply, the taxpayer with the highest adjusted gross income for such taxable year.
(B)More than 1 parent claiming qualifying child
If the parents claiming any qualifying child do not file a joint return together, such child shall be treated as the qualifying child of—
(i)the parent with whom the child resided for the longest period of time during the taxable year, or
(ii)if the child resides with both parents for the same amount of time during such taxable year, the parent with the highest adjusted gross income.
