The more I think about it, the more I think you should trust your hosting. I would make sure that the hosting has a “skin in the game”: That is, they have enough “profiles” that are considered unreliable, it would be very expensive for them (in lost accounts and sales).
And regardless of whether you think the hosting service is trustworthy, you should have a plan if the target account is compromised. Who you will notify, how you can disable this account, etc.
The only technological solution I can think of is that you log in manually, grab the cookie and provide this cookie script - it protects the password, but the supposedly hostile host can use this cookie for any damage it wanted on the target system using any privileges attached to this cookie, including changing the password. So this is not a solution at all.
Oh, talking about privileges: can a task that you need to automate perform with a target account that has reduced privileges, such as a read-only account, or one that cannot make any changes to its profile? Having only your credentials with a low privilege in the hosting service will reduce your risk (or "exposure" as the polysyllabic crowd likes it).
The previous answer, which turned out to be inoperative, is below the line.
You can encrypt the user ID and password using another password. To run the script, its password must be provided. It uses this password to decrypt the web service username and password. Make sure that the script password is not stored anywhere, but only stored in memory and long enough to decrypt the final user ID and password.
If this is really important, make sure that your connection to run the script is crypto (ssh, ssl, etc.) and make sure that the script uses only https to log in.
This does not make you invulnerable to someone with root privileges in the field (at some point, the user ID and the plaintext password will be in memory and therefore vulnerable), but it does more work for them to get the user ID / password.
Updated: Requiring this to be automated makes the above solution not good.