@Herbie, I do see where you're coming from to some degree. The Old Testament temple shekel was a unit of weight, which could be paid in any form of silver - but such measures all ended up being turned into coins eventually for practical reasons. By the time of Christ, the temple had adopted the use of the shekel of Tyre, which had an image of a pagan god on it. This was because the Romans forbade the Jews from printing their own money, and the shekel of Tyre was the closest coin available to the temple shekel (it was actually slightly larger than the historical shekel, so the temple did quite well out of that arrangement). So to pay the temple tax, at the time of Christ, you would have to handle a coin with a pagan image on it.
Roman coins were not used to pay the temple tax. Rather, shekels of Tyre were used. This is why there were money changers in the temple courtyard - people would exchange Roman and other money for silver shekels, in order to use these to pay the temple tax.
So if it was wrong to handle such a coin, then it would be wrong to pay the temple tax.
Where your argument falls over though is that there is no actual statement in scripture that it is wrong to handle such money. Your position is based on speculative logic. And given how common money is, affecting so much of life, you would expect somebody somewhere would have said something clear about this issue.
The temple tax was being paid in pagan coins - did Jesus ever tell anyone "don't pay the temple tax"? The Roman taxes were being paid in pagan coins - did Jesus ever tell anyone "don't use Roman coins, don't pay Roman taxes"? No. Never. Quite the opposite - He told people TO pay the Roman taxes - and they could only do that by handling Roman coins.
If it were sin to handle such coins then Jesus would have been instructing everybody to sin. God would never command sin, so this would mean Jesus was a false prophet.