If Constantine truly worshipped Sol Invictus, even portraying the icon on his coinage, why does the Church insist that he converted?

First, you need to fully understand Constantine, his family life, and get a good view of his entire life.

Did he worship the Sun at some point? Probably.
Having said this, the exact same logic used to call his Christian faith in question could be used against his being a worshipper of the Sun. He was not consistent, therefore it could easily be argued that he was NOT a true Sun worshipper.

The leaders of the Church during his day did not question his conversion. There may have been some who questioned it, but some (like those who held a strict view of baptism, rebaptism, and lapsing during persecution) questioned the faith of many Christians. There are examples of bishops who excommunicate each other, or at least push for this to be done. Lactantius, a fairly strict Christian leader and historian of the period, had a high view of Constantine as both a Christian and a leader.

You can read more about Constantine and the Sun here:
http://www.churchhistory101.com/century4-p6.php
and here: http://www.churchhistory101.com/feedback/sunday-worship.php

Christians of his day appeared to believe that Constantine was legitimately saved.
They considered that:
- his father generally did not persecute Christians,
- his mother apparently was a believer,
- he credited the Christian God for his military victory and ascension to the throne,
- he passed a law to forbid Romans from being persecuted for their religious faith,
- and a law restricting businesses on Sunday, giving freedom for worship attendance,
- he returned buildings/properties to churches which had been confiscated,
- he gave money to rebuild churches which had been burned during persecution,
- he wrote letters calling bishops to prayer/unity in the name of the blessed Savior,
- he showed tender mercy to maimed bishops of persecution at the Council of Nicea,
- he called for Christian baptism on his death bed
(having not been baptized for fear that his oversight of capital punishment might be held against him) and many other smaller things, signs, or fruits of being a Christian were present in his life.

Secondly, the usage of the Sun on coins and other emblems was so common - very much like the American eagle to represent freedom. The Sun represented a form of monotheism in the Greco-Roman world. Having the Sun on a coin would have been similar to having "In God We Trust" on our coins/dollars. There are probably some who would argue that using US money is idolatry.

So how strict would we be to determine that Constantine "truly worshiped" the Sun?
I just returned from Ecuador where fellow Christian believers live in crude houses that many Christians would not want to use for an animal. When I consider our wealthy lifestyles - houses, cars, pets, iPhones, iPods, and Raybans - many committed Christians could be accused of bowing down to the idol of materialism and hedonism.
Am I trying to excuse Constantine? Well, yes and no.
What I am trying to do more than this is to suggest that we should avoid a knee jerk judgement against a man living 1700 years ago in a far away land and culture.
You can read more about pagan influences in early Christianity here:
http://www.churchhistory101.com/feedback/pagan-influence.php

Lastly, we should realize that there is no clear interpretation of exactly what holiness is, or what the requirements are for obedience and holiness.
Every age and every culture has it's own set of requirements.
The tendency for every group, every "school" of thought, is to use their particular set of rules and apply them to everyone else. This is tempting, but is not a good way to read and understand history.

If we judge Constantine for Sun worship:
What do we do with Martin Luther and his anti-semitism?
What about John Knox and how he either encouraged or looked the other way while Protestants beat, whipped, and killed Catholics in St Andrews?
What about John Calvin and the brutal ways he and his flock executed "justice" in Geneva, even killing Michael Servetus who was critical of the doctrine of the trinity?
What about Dietrich Bonhoeffer who engaged in a plot to assasinate Hitler?

You can see the difficulty.
Read more about the early church fathers and holiness here:
http://www.churchhistory101.com/feedback/fathers-holiness.php

Al B.