If you don't filter, you will have lots of solids in your finished back sweetend cider (btw, how are you controling yeast growth with the new sugar add?)
Coffee filter could work, but expect it to take some time unless you have a pump for the filtering. I cold brew coffee and it seems to take forever to drip through.
Personally I'd filter (which will take along time exposed to air) and then add something like campden or such to killl wild bacteria and freeze - with the jug inverted which is tricky because the ice will expand. If glass possible break, if plastic, it might knock out the cap. But upside down is better because the water will push the sugars down as they freeze (think kool-aid ice pops)
After freezing, I'd invert the jug and let thaw watching the frozen cider to see if there was any color left in the remaining ice block. The non water parts thaw first, but still some of it could be trapped up in the block. Up to you how badly you want to get those last bits.
I think if you let it thaw about half way first, the results won't be as good.