Lebesgue measurable but not Borel measurable
Solution 1:
Bit of a spoiler: Your approach seems on the way to what I've seen done, but instead of trying to intersect your set, you might want to map a non measurable one into it using a measurable map and remember how preimages of borel sets behave.
Spoiler: your map could be one from the unit interval onto that very famous set by that very famous guy born in 1845 who suffered from depression and the dislike of many of his contemporaries... ;-)
Solution 2:
There is some good stuff here:
https://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-choice