I tried the command John suggested to set the password through the command line. Since my application pool has spaces in the name, I got this error message:
Failed to process input: The parameter 'Sites'].processModel.identityType:SpecificUser' must begin with a / or - (HRESULT=80070057).
I was able to set the password with the following command:
%windir%\system32\inetsrv\appcmd.exe set config "/section:applicationPools /[name='APP_POOL_NAME'].processModel.identityType:SpecificUser" "/[name='APP_POOL_NAME'].processModel.userName:DOMAIN\USER_NAME" "/[name='APP_POOL_NAME'].processModel.password:PASSWORD"