Pay for a 1 month subscription to Seats.aero, put in your dates and cities and then hope availability shows up for a transfer partner from Chase.
United doesn't seem to fly direct from IAH to CDG or FCO, so you'd probably have to fly on a foreign carrier like Air France. You can transfer points to Flying Blue and book a business class ticket for 50k each way if there is availability on your dates from IAH-CDG.
Emirates operates a direct flight from IAH to Manchester England and they have a daily flight from JFK to Milan that you can book on points transferred from Chase, but neither of these get you where you want to go exactly without transferring planes.
Air Canada Aeroplan is a Chase transfer partner and you might get lucky with award availability on Turkish. I believe their flights to/from Europe can be as little as 45k each way in business. The caveat is you have to connect through Istanbul.
You can get to Rome on ITA Airways by transferring Chase points to Virgin and booking through them as a partner award. ITA doesn't fly direct to IAH but you could reposition to JFK, Chicago, Boston, etc.
Some airlines will let you buy economy and then upgrade to business later but I'm not familiar with which ones have which rules.
September might be slim pickings for award flights, but December if you jump on it should have a lot more availability right now since schedules would have just opened up.
For reference, for my honeymoon in September we are flying to JFK to reposition to fly Emirates Business class to Milan booked on points (87k per person, one-way).
We are flying back from Rome to Boston in business class on ITA booked through Virgin on Chase points (75k per person, one-way). All in we used 324k points that we had saved up (Actually a little less, since at the time of booking both Emirates and Virgin had transfer bonuses). We just had to come out of pocket for our positioning flights to New York and Boston, which can be pretty cheap if you're flying Southwest.
ETA: When I'm searching for routes I like to use a combination of googling "____ Route Map" and clicking whatever result pulls up the flightconnections.com website to see where all I can go direct. So if you google "IAH Route Map" you'll see where all in Europe you can fly without connecting. You can do the reverse by searching like "FCO Route Map" and seeing which cities in the US might be easiest to reposition to for better award availability.