in the movie the Green Knight, Gawain receives a pagan and Christian blessing, and receives a pagan magic belt, without any controversy. Was Christian and pagan syncretism common at the time the original epic poem?

Not a historian but a medieval literature postgrad so I'll give it a go.

The magic belt is a plot device in the original poem - granted by Sir Bertilak and really the centrepoint of the plot. To summarise very briefly, Sir Bertilak is the green knight; he sets up the beheading game and the sharing game with Gawain; he has his wife offer Gawain a girdle that shall protect him from harm; by accepting this Gawain wins the beheading game (as he can't be hurt) but loses the sharing game (as he accepted something without sharing it with Bertilak)

Now, this belt in the source isn't exactly pagan. Bertilak tells us:

"Thurgh myght of Morgne la Faye, that in my house lenges,/and koyntyse of clergye, bi crafte well learned./the maystrès of Merlyn mony has ho taken,/for ho has dalt drwry ful dere sumtyme/with that conable Klerk, that knowes alle your knyghtes at hame./Morgyne the goddess/therefore hit is her name;/ welded non so hyghe hawtesse/that ho ne con make ful tame"

I don't have a translation to hand so here's my quick approximation:

'through the might of Morgan the fey, who is staying in my house, and who is well skilled and learned in magic arts. She has learned many masteries of Merlin, for she has dallied with that knowledgeable clerk, as all your knights at home know. Morgan the goddess therefore is her name: no one has such great pride [lit. Haughtiness] that she cannot make them tame'

Morgan is also behind disguising Bertilak as the green knight. In most versions of the mythos Morgan gains her magic powers by one of two means: either by seducing Merlin, as here, or while at school in a nunnery. Merlin's knowledge and power descends from both satanic and divine forces (he's the son of an incubus but due to his mother's holiness god grants him leave to use his power) while it is necromantic books from the nunnery that Morgan uses to teach herself.

So Christian and Pagan syncretism - undoubtedly there is supernatural working in the narrative, but this is entirely explained by the Christian worldview. Magic isn't coming from pagan gods or arcane rights - it comes from god, a devil, or secret knowledge of how god's world works. Many Pagan elements of Arthuriana are drawn into christian versions. For an example featuring Gawain, early stories had him blessed by the fey folk at birth to increase in strength until noon, then dwindle after. Later versions tend to either mention this blessing but not the fact that it came from fairies, or just gloss over the element entirely.

So why does the film up the Pagan elements? Well for one it's trendy. Neo-paganism and neo-celticism are pretty modern cool topics due to a variety of reasons around religious freedoms, anti-colonialism, and Christianity generally being uncool. It helps that GatGK has a long history of pagan readings, largely due to the parallels between the green knight and the green man of British folklore (incidentally one of the professors at my uni said multiple times that he wants to fail all undergraduate essays discussing this because he reads about a dozen a year and none of them have any real point because we have no surviving record of folk stories besides the ones collated from oral histories around the C18. Real historical paganism is basically a dark topic in history because records are so scant or so obviously biased)

One must remember that medieval society had a complex relationship with magic. According to the Catholic church magic did not exist - anything seeming magic was an act of god. However, things like alchemy, astronomy, holy geometry etc were believed to be scientific endeavours because they were working out the truth behind the workings of the world (with varying levels of success...). In Arthurian literature, this is usually brought in as a mechanism for dramatic purposes. Merlin isn't very interesting if he's just a smart guy, so his magic comes from a daemon via God. Morgan could be just a scheming alchemist, but it's more interesting if she can do glamours and spells so let's make her a necromancer.

So, was christian and pagan syncretism a thing at the time of the poem's composition? Well, sort of, but sort of not. It didn't exist in the sense that Mozarabism was a Christian/Islamic syncretism in Iberia, but medieval Christianity undoubtedly painted itself over pagan magical tropes that existed in Britain's mystical past. Morgan still has the epithet 'the fay' even though very few texts give her any link (or acknowledge the existence of) to the fey folk - the fairies might be gone, but the world implies magic so they keep using it. Nowadays paganism, fairies, religious syncretism etc are all more popular, so it seems the filmmakers have played up this element for artistic reasons.

J.J. Anderson, Ed., Gawain and the Green Knight, Pearl, Cleanness, Patience (Everyman, 1996)

Hope that helps!