I think 巴黎买 carnet is better than day pass, buy their

museum pass will save you from waiting in line (and money depends on how many museums you plan to go)

In Rome, Roma Pass worked great for us, it let you in 2 museums (梵蒂冈 is not included though) + 3 day metro usage.

Personally I will stick to one country instead of rushing to another. You can see more details in my blog.

have fun