There are three options which may or may not work well for you:
You keep your US phone back home in the US and try to have SMS messages forwarded while travelling. You will need to make sure your phone stays connected while you are away, though that shouldn't be too difficult.
In order to have SMS forwarded somehow, either your operator has an option to do so. I understand Verizon does, but it works only partially.
So you might try and put an app on your phone which will forward incoming SMS messages to email or any other phone. Note the difference between your phone doing the forwarding or your operator doing a cc of any of your SMS for you to some app over the Internet.
N.B.: It might be by full purpose that certain types of SMS are not forwarded to the app on your PC for security reasons.
What will make your life difficult is that Google decided pretty recently to ban a lot of SMS forwarder apps from their Playstore. Again as a mean to prevent fraud and enhace security. This will lead to the fact that you may find lots of "how to forward SMS" articles on the Internet, just you will not find the apps mentioned anymore.
There are some still / again available but I found many of them not quite fit for the job. For example, some of them don't work well with your phone's power management. You unused phone will go to sleep at some point and will not get waked up far enough for the forwarding app to forward the SMS once it arrives. Some apps need 10-15 minutes to forward, which is next to useless for SMS that are confirmations which are valid only for 5 minutes.
Your milage may vary ... The problem is that by the time you may find out it doesn't work as expected there is literally an ocean between your and your US phone.
Take your US phone (or SIM) with you
In a comment you say Verizon will charge you 10 USD a day if you roam. But I understand this will be the case if you want to use your minutes and US data in Europe, which is not at all what you want. You just want to receive SMS messages.
So do not buy any "traveller plan" or anything but just make sure roaming as such is enabled on your US sim card in case it isn't by default. Make sure you do not switch on data roaming on your US phone while in Europe. I assume you will not incurr any charges than but you will receive your SMS messages.
Buying any non-smart cheap GSM phone to put your US SIM in (like a Nokia 105 or 3310 for example) and you have a perfect SMS receiver device and can use your french SIM in your Android phone.