my guess is you wont be able to do this across versions ( not 100% sure on that ). you could change your submit type to use the list web service of the 2nd farm, that should work (depending on authentication configuration - read more below).
I think if you looked at Fiddler you will see authentication errors, I didn't think about it initially, When saving directly to SharePoint Library in 2010, you are probably passing a claim from 2013 to 2010 (which I am going to assume is NTLM). Even if 2010 was configured with Claims you may have issues because it is not going to trust claims from a 2013 farm unless you explicitly configure it to do so.
With web services, you need to call the service with the domain credentials that have access. WCF is not Claims aware so it wont know what to do with the identity it is presented. You can try and utilize a "web service proxy" to get this to work with InfoPath. I'm not entirely sure it will work with the 2010 web services.
- Create a specific data connection file created from InfoPath
- Enable data connection authentication for the form.
In both cases you need to have the Secure Store Service running and create a target application that uses windows credentials that can access the 2010 farm.
This enables InfoPath to use credentials stored in a connection file.
Set-SPInfoPathFormsService -AllowUdcAuthenticationForDataConnections $true
In your InfoPath form, you can create a connection to your web service and then save it as a connection file. You will have to open the file and modify the XML directly:
useFormsServiceProxy = True
<udc:Authentication><udc:SSO AppId="" CredentialType=""/></udc:Authentication>
AppId will be your SSS Target Application ID. You will need to upload the connection file to your connection library and reference it from InfoPath.
Full disclosure is that I don't spend a lot of time with Web Services. Here are some Microsoft References you may want to review.