There's not much you can do. If you did (for example) claim your children as dependents, you would reduce the tax you owe but you would increase the tax your partner owes. If your partner's income is substantially higher than yours, their tax might go up by more than yours goes down.
It's not clear to me that you partner's employer would be required to penalize them if they enrolled in insurance coverage for the children but didn't claim them as dependents. Your partner's biological children should be eligible for insurance coverage even if they are not claimed as dependents.
Remember that withholding is only an estimate of the taxes you owe, your actual tax bill is calculated on your tax return. If you are over-withheld, you get the difference back in your refund. 40% seems high, that could be 22% federal (if you are single making more than about $50,000), plus 7.65% medicare and social security (everyone pays that, you can't do anything about it), and 10% state tax? Sounds a bit high but could be in California I suppose.
You can prepare different test tax returns. In general, if the partner who pays more than half the household expenses claims at least one child, that partner can use head of household filing status, which has lower rates than single. But only one of you can be head of household, the other must use single status.
If you pay for child care so you can both work, there is a credit for that, which is a bit larger for lower income taxpayers, but not by much. The lower income taxpayer might qualify for EIC (but if the lower income taxpayer claims both children the higher income taxpayer loses the ability to claim head of household.)
There's no right answer, you need a tax professional who can review your confidential details with you or you can prepare different tax returns using different combinations of situations.
