The heliocentric model already had support long before the time period you are after (1200s). In fact, some of the Greeks tried to prove the heliocentric model thousands of years ago. There was also religious support for the heliocentric model even before that.
There were some bright Greek astronomers who were on the right track and viewed the cosmos similarly to how we do now. They realized that if Earth is orbiting around the sun, then there should be a calculable parallax for the stars. That is, if you make astronomical observations at one point in the year, then you make the same observations precisely 6 months later - when we would be on the opposite side of the sun - we should have moved enough to see some of the stars in the sky change location relative to each other.
The calculations are simple trigonometry. The Greeks tried it, and calculating the trigonometry would have provided difficult to dispute proof of the heliocentric model.
Interesting side-note: According to the Wikipedia page on trigonometry, "The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies". The study of triangles started long before that; this is referring to trig much more as we know it today. So it was the astronomical observations and debates much like the one that we are discussing in this Q&A that actually established current trigonometry.
The problem, which the Greeks were unaware of, was that even the closest stars are so unreasonably far away that the parallax is so small that you need a descent telescope just to measure it. The Greeks were so close to proving the heliocentric model, but their astronomical observations just did not provide enough precision to discern the parallax. And so the geocentric model had a victory instead of the heliocentric model.
So the answer to your question here is to provide the Greeks with better astronomy equipment. That shouldn't be too difficult. Just shift the timeline of the telescope forward by giving them an improved control over optics.
Citation for the stellar parallax, including the Greek use of it.
According to the above source, from Penn State, the stellar parallax observation was not successfully done until the 1800s. So you might need to shift the development of telescopes forward a lot. Though usually developments take many generations, it is not unheard of to make drastic leaps and bounds in 1 generation if you get a dedicated individual who devotes their life to it, such as was done for clocks when moving from weighted systems to springs. You could have a history where a Greek person obsessed with optics spends his life pushing the telescope hard shortly before the heliocentric debate happens.
Religion: The Diplomacy
(Edit: See new note at bottom of this section)
Religion itself was not a barrier to the heliocentric model. At least, the predominant Abrahamic religions (Judaism, Islam, Christianity) were not; I cannot say with confidence about other religions.
If you are thinking about the catholic church's blocking of the heliocentric model, considering it heresy, that was not a religious issue. That was purely a diplomatic issue. The catholic church was a governing structure which relied on its word being unquestioningly accepted. If the catholic leadership allowed their word to be questioned, they lost their power. The catholic church's actions should be viewed as the actions of a government, not a result of religion.
In fact, ancient texts used by the Jews, Christians, and I think the Muslims too actually support the heliocentric model. Despite the catholic church's own proclamations otherwise (which was essentially heresy by the church government), their bible actually states that the Earth is what moves through space. Earth is said by the bible to be a sphere that travels through space, and yet many of the governing leaders who abused the faith of their followers tried for centuries to tell us that the Earth was flat or that it was the stationary center of the universe, despite that bible.
So religion itself actually supports the heliocentric model, but the people did not accept that.
So the answer to your question as far as religion is concerned is to change the people somehow so that they are not arrogant jerks (not the words I want to use, but I'm keeping it G-rated). Let religious people actually practice their religion and develop it instead of having dictators pervert religion to their whims. Wouldn't that be grand? In theory, this is actually a smaller and easier change than the one above about changing the history of optics, but in practice changing people to be better is very difficult.
Please note: I am trying to find sources for my claims in this section. It is easy to find the claims in the bible that Earth is round and suspended in space, but I am having difficulty finding the one I recall about it actually moving through space. Also, some people are pointing out in comment the supposed biblical evidence for flat earth and geocentric models: to that, I suggest people take notice of the fact that the sections that sound geocentric are often in symbolic, metaphoric sections (ex: Psalms is literally a collection of songs and poems), and I find such biblical geocentric claims to be weak personally - but yes, there are hints of geocentric ideas as well. I would also point out that the biblical heliocentric support is generally in the more literal sections. Overall, feel free to take the biblical claims with a grain of salt until such time as I update this section; however, the claim still stands that the catholic blockade of the heliocentric model was, as many catholic actions through history, a diplomatic power play rather than a religious requirement. This is evident in that many supporters of the heliocentric model were themselves very religious Christians as well.
Resistance to Change
Your other issue was the generic resistance to change that many people have.
There is not a lot you can do about this. However, given one or both of the above changes, I don't think you would actually need to do anything else special here.
If the religious texts were actually taken at face value, instead of having governmental perverts keeping people from reading their own religious texts, then the religious groups would have been on board with the heliocentric model.
If the Greeks had made better observations, they would have been on board with the heliocentric model.
The Greeks and the Abrahamic religions were, arguably, the two most influential things in history to much of science and art that would follow, even to this day. If the people in both those camps supported the heliocentric model, I don't think there even would have been a change necessary, as it would have been the default.