Caesar's Image on that coin might not have been their main issue, in actuality. Here's why:
To begin with one representative sample of Pharisaism, the historical Hillel school dates from an era when Julius' Caesar's famous tax exemption still applied in Judea - that Jews would pay no tribute in the seventh year. While King Herod seems to have challenged that custom, and his son Archelaus infamously gathered ten harvests in ten seasons, the direct Roman poll tax did not come into Judean experience until the registration of Quirinius.
At any rate, before the year 6 AD, Pharisees who were sticklers about the Mosaic Sabbatical custom had an easy scapegoat for that custom's latest demise - it was all because of the Herods. However, in 6 AD, the newly established Government under Quirinius introduced a new poll tax (tributum capitis) which, distinct from the land tax (tributum soli), was not agriculturally based. But one thing the poll tax was, obviously, was perennial.
Whatever else changed about Judea's tax situation, in this sudden transition to direct Roman rule, the poll tax (by itself) was probably seen by many Pharisees as the first Roman reversal of Caesar's famous decree. And in that, I suspect, certain Pharisees may have taken the poll tax as an affront in itself, in all years, from the standpoint that - from where they sat - this new tax seemed deliberately contrived as a method of sidestepping controversy about Judea's traditional sabbatical laws.
One caveat: I'm not certain Quirinius actually instituted the land tax, or that one had yet been levied on Judea to this point. But even if it had, any Pharisee who wanted to be strict about the year of rest could store up his own grain supply for six years. As long as he was not a farmer, then (1) the debate was largely on principle, if not merely academic, (2) the land tax would not affect him directly, and (3) the poll tax would still come to him in the seventh year, and thus could still feel like an affront.
Of course, I'm not suggesting all Pharisees were concerned about this, and I'm sure that on a more common level of outrage the Image of Tiberius on that coin had to be the most [or, rather, least] popular sticking point with any traveling Rabbi's constituency. I admit, that aspect of the poll-tax issue probably had become a very common complaint, but was it really a significant problem? There's no record of grass roots non-payment across Judea, and anyway, practically speaking, isn't that why Judea had money changers?
Point: As long as Roman tax collectors took shekels, a Pharisaic complaint about Roman imprinting seems more like empty rhetoric. The more I think about it, empty rhetoric may really be all that it was.
We now have to consider that maybe this trap-question, as laid out in Mt.22, Mk.12 & Lk.20 (but also assuming historicity, here) was nothing more than a really weak effort to trap Jesus, which then turned into a good story for the Gospel writers, largely because it makes Jesus' response look more brilliant by comparison. On balance, all of that seems quite plausible; but there's another side to this *coin*, also, which requires more thought.
What if, instead, we consider that maybe the trap set for Jesus on that day invited him into a slightly more complicated debate, one involving more than just moral shagrin for the imprinted face of Tiberius, on a coin no Judean was ever required to touch. It should occur to us that - even though the graven image aspect may have played best with the crowds, and may still be what plays best in repeating the story - a more entrenched kind of dispute appears to be what we ought to expect from the larger historical context. That is:
Both Mark and Matthew* frame the trap as a "choose sides" demand, as if the Pharisees and Herodians have been fiercely debating the poll tax issue for quite some time. However multi-faceted that debate might have been, I don't see the Herodians giving it much time or respect if the Pharisees' major case could have been overturned with the quick response, "Well, they take shekels, don't they?"
To modern readings, the Gospel writers seem to suggest the main coin issue was about graven images. But our 'impression' here may only be due to the fact that we've lost hold of what everyone knew, back then. When Mark and Matthew say, "Pharisees", "Herodians" and "κῆνσον" (poll tax) in the same sentence, I suspect there were some in their original audience who knew much better to what larger political conflict those three elements alluded. So, each writer wisely included that data, but overall managed to play to the crowd. Just like the Pharisees did... and just like Jesus, as well.
By the way, just to clear up one side point, from above: Jesus' response in the Gospels is most brilliant NOT because it overturns the tax trap so effectively, but because the Lord steered his way through a divisive contention by turning it into a challenge for all souls to acknowledge the centrality of God.
Because, naturally, we all have been formed in the image of God.
*I don't want to distract from this post here, but I find it characteristic for Luke-Acts to avoid repeating negative views of the Herodians, as often as possible. I've always suspected this had something to do with Agrippa & Bernice being on Paul's side before his voyage to Nero, but I've not worked through that arugment yet at all. Something else someone (else) should work on, someday. (!)