First, welcome to the ACA Forums.
A lot of the bike shipping companies have recently greatly increased their shipping costs. I would suggest you check the costs to ship via the various methods and then compare to what it costs to take on a plane/train. I am going to Europe this summer and the cost to ship a bike via BikeFlights was around $700US each way when I checked back in February. I plan to pack the bike carefully and take with me on the plane ($100) unless the prices for BikeFlight an the others have dropped dramatically.
I am unfamiliar with the Spokes bike shop so can't help you there but it seems to be a good shop based on its Google reviews. I personally would probably take the train from Constanta to Bucharest as you most likely do not need to disassemble your bike though Romania is a bit different at times when it comes to trains. Europe as a rule is very bike friendly, at least compared to the US.
For a route between Bucharest and Constanta, you might check out the heat map at
https://www.strava.com/heatmap#9.57/27.43456/44.17572/hot/ride . The brighter roads show what is popular with the local cyclists. Remember though that European riders typically have a higher tolerance for traffic than North Americans so a popular (bright) road might have more traffic than you might like. Use the Streetview on Google's Maps to spot check the route. You might also ask some bike shops in Bucharest for the best route on how to get out of Buchrest and Spokes on how to get into Constana. The area in between is not as populated so the routing might be easier.
Also, you might ask this same question over at BikeForums.net as their touring sub-forum is a more international audience than here. You could also look at the journals on CrazyGuyonabike.com . Here is a list of journals with the keyword "Constanta" in them:
https://www.crazyguyonabike.com/search/?main_type=journals&docs=all&query=constanta . You could also search on other keywords like EV6.
I am unclear what you mean
"does anyone have any suggestions for a EV6 trip that starts on the Black Sea and heads to France??" since that is exactly what EV6 is other than most people go W>E.
Finally, you might consider editing your post and removing your email addresses since you might get a lot of spam as a result of it. I don't know how it happens but it does. People can private message via this website (see the link under your user name) if needed.
Hope you have a wonderful trip! Tailwinds, John
