You are already using great protocol for remote access (SSH). I would recommend you to use it for running commands remotely too. PuTTY you mentioned contains also a text console SSH client
plink. If the remote scripts are not very interactive the best option would be to use
plink [options] [user@]host [command]
If the script is highly interactive it could be better to use the classical GUI PuTTY.
putty -ssh -l user -pw password -m command.txt host
In this case you have to create a file (
command.txt) and store the command to execute there. Here is more information about PuTTY command line.
For this use of SSH it could be very helpful to use a public key authentication instead of passwords. Then it is possible to run the commands without entering a password. See for example: Public Key Authentication With PuTTY or Key-Based SSH Logins With PuTTY.