I book 4xJ awards at least twice per year. It takes a lot of searching and flexibility, but it can be done.

For example, just booked our vacation for next summer LAX-DFW-CDG on AA via AA miles (56k miles per person) late July and return BRU-CDG-LAX mid August on AF via Amex transfer to Flying Blue (47k miles/per person + fees around $400 per person). The two trips we took this year (booked last year at 11 months out) we flew ORD-FRA-VIE on LH booked mostly with existing UA miles + Chase transfer (88k miles per person which is not ideal, but we were specifically trying to fly on a LH747) and MUC-MAD-LAX on IB booked via BA Avios via Amex transfer with bonus (effectively 72k miles per person) in July/August and LAX-CDG-LIS on AF booked via Chase transfer w/ bonus (effectively 50k miles per person + fees) and TLS-LHR-LAX on BA booked via Alaska MileagePlan (65k miles per person + BA fees -- my guess it would probably have been 90k/person if booked with Avios via Amex transfer). So I have spent a fair amount of time searching for 4 business awards US west coast to Europe.

For Air France, I have found that for the lowest awards you usually need a connection via CDG. For example SEA-CDG will probably be many more miles than SEA-CDG-XXX. Since you are trying to get to Italy would check those connections first but if they price high look at NCE or MRS as the final destination. I found connections to other French airports or west (Portugal/Spain) for the lower amounts vs. trying to connect into Germany or points east. AF also has codeshares where the TGV train is the intra-france leg - sometimes that kicks out much lower rewards or more available seats. My return next summer is actually ZYR-CDG-LAX (Brussels train station as the starting point). Perhaps plugging in a few TGV end points like Lyon that get you towards Italy is worth playing around with to see what pops up. That said if you can get to Paris there are many LCC options to get you from there to Italy (unless you are set on business all the way).

Similar strategy for Star Alliance partners you could book via AC. I booked via UA on LH but the same awards were visible on AC -- ORD-FRA was really high but ORD-LHR-VIE (same ORD-FRA flight) was at the saver level.

BA also often has 4 J seats available on many days/routes -- you just have to view it more as discounted business class vs. a true award with their additional fees. The fees on a return connection to the US from continental Europe via LHR are better than the way there.

The other thought is to look at availability from Toronto/Montreal or US East Coast -- book the business award from there -- and buy your positioning flight(s) for this side of the pond.