Very tentatively: paying taxes, properly so called, must be different from other forms of cooperation because, as you say, Our Lord did it. Taxes go into a general government treasury. Once the funds are there, you have done your duty (rendering unto Caesar that which is Caesar's). Then it is Caesar who does evil if, in a completely separate act, Caesar decides to purchase incense for Jupier, augment the crucifixion program, or invade Gaul. But that would be Caesar's sin, not yours.
On the other hand, if Caesar were demanding funds that did NOT go into the general government treasury, but which specificlaly went into the "Incense-for-Jupiter" or "Crosses-for-Jews" fund, then you would be morally obliged to disobey, because its use would be already explicitly specified.
By analogy, you could pay Income Taxes, which go into the U.S. Treasury general fund. But it would be morally wrong to pay funds into a payment scheme designed as an "employee benefit" (insurance plan) when YOU KNOW the "benefit" consists of injecting toxins into employees or providing for their reproductive maiming.