Is it permissible to open a store in the Philippines with a Christian partner whose money is halal, and for Christian women who are immodestly dressed to be in charge of sales?

If the store's activity is permissible and its transactions are regulated by Islamic guidelines, so that forbidden things like usury, selling alcohol, and pork are avoided, then there is no harm in participating in it, though it is preferable not to. You must also not allow the sale of unveiled Muslim or non-Muslim women in it.