While publishing database, we need to cleanup the dev configurations a little further.
In the connection string check what is the Database server name on your client hosting environment and include that in: Data Source="Client database host server name"
Also instead of providing .mdf file ( as specified in your connection string with AttachDbFilename='C:\Users...'), better would be either generate the script from SQL Management studio in your local and run those in your client environment.
If it is by requirement to provide .mdf file then make sure the path to mdf file in hosting environment matches to what you give in your connection string.
Or else you can provide mdf file to client ask them to attach it to the hosted database server.
Often, Integrated Security might not be provided on host servers. So check the userid/password with your client on hosting machine to connect and update them in your connection string.
Hope some of these point might help you.