- Your god wants this to be a test, then Make it a Test.
95% of people with 95% certainty gives us a little bit of wiggle room. We are going to lean on that, hard. Also, we should blame the boojum (it will make sense later).
So, pick a semi-plausible explanation. It can include any of the other answers, it can be something else entirely - this answer is about method. I'm partial to the idea that some kind of low level reality-warping ability in the population - the weather shifts, the fires come, the shield manifests, maybe the warrior comes - because people believe it does, and if enough people believe with enough power, the low-level ability from each person works together to manifest it. Those particular usages are often seen because people believe them, heard of them in stories, and as they manifest, more people believe - so they show up all the time. Anyway, your god now has an explanation in hand.
- Now, to make people believe it, subvert your leaders.
Pick your most influential religious leaders, your top political leaders, your little local pillars of the community known for being the most faithful and devoted followers - not more than 1% of your total population of each, maybe aim for just under 2-2.5% total instead, since you want to keep as much wiggle room as you really can. You are ranking people by how many others look up to them, how faithful and devout they are perceived by the community, how influential they are.
Have your god go to them - yes, this means they won't disbelieve in him, but if your god chose wisely, they wouldn't have anyway - have your god go to them and set them each a test of personal devotion: that they must not publicly support your god for a year, even publicly deny him. They may not explain or discuss their reasons until the year is up. If your god wants to be a little more sympathetic, would not like to be seen as insecure, or would like to eventually be worshiped again, have him blame the boojum.
- Step three in your plan, have the "explanation" discovered
So, your god has set a test for just under 2, or 2.5, or even 3% of your society. The reason for "just under" your chosen percentage, is so that you can grab some scientists. You will need only a few, but want more than a handful - preferably widely spread, well respected, and quite devoted. They need to be convinced of your theory, so that it can spread and be believed by your general population. Come up with a few tests that "prove" your original explanation, like a psychology test ends up letting groups of people recreate the manifestations if they believe hard enough, or certain rituals will let someone control the weather, or these special rocks will let one influence the fire, or whatever proof your god came up with to fit the explanation. Some of your scientists (up to the percentage) will be lead to the experiments first (by influence direct or indirect, maybe lean on your leaders and priests), others will need direct visitation. A number of top scientists "discovering" the phenomenon "independently" will give it legitimacy - especially if some are reacting in genuine horror at the implications. It's important that the test, whatever it is, is actually reproducible, at least for your one year, as that will be the "proof". Remember to blame the boojum, your scientists will be a lot less forgiving of a deliberate crisis of faith if they don't think there's a reason!
- Your primed percentage will now Stir the Pot
Scientists declare breaking news, warn the town criers! The scientists start spreading the news about the test, and its possible religious implications - some in horror, some in denial, and some - because their beloved god asked it of them. The ordinary people look to their religious leadership for a response, for acceptance or rejection of this test and its implications. The top ranks, and the most devout and influential - are silent. The political leadership as well. All the top ranks of the people most likely to decry the implications are silent, and some even go so far as to deny your god, or reject him publicly!
Your common people are shocked and appalled! What the hex-and-a-quarter is this! It seems to them like even, or especially, the most religious people have had their faith shaken or even broken by this test. Laymen look to their priests, priests look to their leaders, communities look to their local faithful, and they make no effort to encourage or support faith in your god, or even talk about the reasons why not (which looks like personal crisis from the outside). Individuals might have laughed of the test, or failed to believe its implications, or just not cared - but this apparent crumbling of the most steadfast, of those they look up to and who inspire them, will shake their own faith, and fast.
- As people's faith is shaken, Watch for Resistance
The lack of leadership will do a lot to shake people's faith, but it is, after all, only two or three percent of your total population at most. Some will be convinced by the test, some more convinced by proxy - but in a population of devout believers, you will be seeing push-back fairly soon, as those devout individuals who remain start trying to rally people's beliefs back to the faith. So your god will need to wait, and watch, and look for focal people and places of resistance. Your god will want a budget - for the whole year, mind - of no more than 1% of the population (this brings your total up to 3-4%, if you're mathing). Knots of resistance can be identified, and their development monitored, by your god and/or those followers in the know.
After they have had some time to brew, the better to polarize people - you god will pick among the top leaders (looking for the minimum of those whose faith is the most trusted and leaned on), and repeat the first trick - visitation from the god they are so devoted to, asking them not to support him publicly at this time, as a test of faith - because of the boojum, of course. The loss of the top people (and again, the most trusted rather than the official leaders) will undermine the resisting group a lot - and being let down or betrayed again and again will badly shake up the faith of those who do still believe, as those who seemed to have the most personal faith seem to loose it, have breakdowns, and denounce their god and previous faith.
You have a year of this - and you want people to have as much time as possible, to react to each wave of defections, the better to seriously shake their faith. You want to minimize the number of people your god visits, too - you will want as much wiggle room as you can for people who won't lose faith naturally, and you have already shaved that thin with your visitation people. Honestly, I would be surprised if you couldn't get a pretty decent majority seriously doubting the god, with shenanigans like this, and perhaps your 95% will be willing to profess disbelief by the time your year is over. Maybe that's enough, maybe not - but it is about all I've got.
- Ah, wait-a-sec. This was a Test of Faith, remember?
So, go back to your 3-4%. Your god talked to them at the beginning of the year, recall, and not only asked them to not support or deny him publicly, but also not to speak of their reasons why - without telling most of them what else was going to be happening this year. Why does it matter? because this is also a test of faith - a fairly good one. These are people who believe, and strongly (your god picked them out of the people who probably would not turn away, recall) - and they have seen their society crumble around them while they had to remain silent. They will be looking at the societal crisis of faith, and thinking what they would have, should be, could be doing to support their students, and friends, and mentors - to do their jobs. To help their people.
It will be very tempting for these people to give in, or give up, and explain to people why they have been silent - that they didn't falter in their belief in the god. Those who manage to keep faith, and follow your god's instructions even when it must have seemed to them as if the end of the world, have passed a pretty strict test even if it isn't the same one. Those who gave in and spoke up, have failed - even though they are professing their faith in the god, they didn't have faith that his instructions were the right path. If your god is willing to accept this alternate, backwards test-of-faith, then between the extra percentages now passing the test, and the extra percentages of wiggle room for those who can't be convinced, I think your society now has a good chance of hitting that 95% population with 95% certainty.
Recall I said back in the beginning, to blame the boojum? Well, you god will look pretty insecure, and possibly cruel, if he's testing people, and shaking their faith, frightening them, and upending their society just for a test. So he can blame the boojum instead. It doesn't matter what the boojum is, or what it does, or even if it is explained or not - ever. The point is, he can say the test has to happen, and it has to happen now, because of the boojum.
He can say, afterwards, that it was necessary for 95% of the population to not believe in the god at the end of the year, or a specific date, to prevent...something... from happening, because of the boojum (if that was achieved). He can say it was important for the whole society's faith to be tested during the year, whatever the actual result, because of the boojum. He can say just the leaders' faith (the percentage people) needed to be tested, and the "test results" happened to happen at the same time as that test - because of the boojum. The boojum can be an actual explanation he has - or he can just say it is an inscrutable godly thing, that can't be explained. All people need to know is that they're safe, the god's testing/meddling/whatever protected them from the dangerous boojum-thing.
So they won't blame him for the Great Crisis of Faith, or for the social unrest, or all of the trouble and fear during the year. So he can have people believe again, by having the percent-people announce their year-long test of faith (and the bargain), and perhaps appearing himself, to explain what happened and why (because of the boojum, weren't you listening), without sounding insecure. So he can meddle again, and not have to uphold the illusion of whatever explanation he concocted (which might have been annoying, being at the back and call of scientists instead of meddling freely).