Train alternative

Try rome2rio.com for options. Looks like renting a car would be simplest, a three-hour drive to CDG, but expensive. There's also bus service, taking longer, without too many stops but mostly involving a change, Don't use r2r for booking, book directly with the bus line. For a car, try Auto Europe, a US-based brokerage that works with numerous rental companies.

or split things up, start by car or bus and change over to plane or train

I don't know too many planes that will get one between Bayeux and Paris ! Well I suppose the wealthy have their own personal jets.

Warning about Bayeux Shuttle. They are in financial difficulties. They canceled on me two days before the tour after confirming the day before.

Fortunately, I was able to book with Overlord Tours at the last minute. It was one of the best tours I've ever taken.

The issue I see for your car rental is that July 3rd is a Sunday. There may not be a location open on Sunday. Buses from Bayeux may be hard to find but there might be more Caen. Also, you may want to consider a private car service. You have a lot of time so maybe contact the Bayeux Tourist Office and see if they have any suggestions or know of a reliable local car service.

Detailed info for track work in July is not available yet. It should become available at the beginning of May. I am a little bit surprised to see no options at all, whether on the direct line to Paris or on the more convoluted routes via Le Mans or Surdon.
I would check tickets again in 2-3 weeks.