[System.Reflection.Assembly]::LoadWithPartialName("System.Threading")
[System.Reflection.Assembly]::LoadWithPartialName("System.Globalization")
[System.Threading.Thread]::CurrentThread.CurrentCulture = [System.Globalization.CultureInfo]::CreateSpecificCulture("en-us")

Add-PSSnapin *exchange*

$start = (get-date).adddays(-3)
$end = (get-date).adddays(-1)

$filter = "(Received -gt '"+$start+"') -and (Received -lt '"+$end+"')"
new-Mailboxexportrequest -mailbox MAILBOX -ContentFilter $filter -filepath "\\PATH\to.pst"

This exports the mailbox from only the day before yesterday.

Sources:
http://rp-it.blogspot.com/2013/05/new-mailboxexportrequest-fails-when.html
http://occasionalutility.blogspot.com/2014/03/everyday-powershell-part-17-using-new.html

Exchange (2013 and other) support sender based routing only through 3rd party software such as described here or here or here. Too complicated or costly I chose to follow this post by using free hMailserver to achieve this:

• fresh install
• no domains setup
  
• Settings > Protocols: check only SMTP
• Settings > Protocols > SMTP > Routes
  #1 Domain: default (this is just for naming purposes)
  Target SMTP host: our.default.mailgateway.xyz
  Connection security: STARTTLS (because why not)
  When sender matches: local
  When recipient matches: remote
  (those two settings should have no effect what so ever)
  
  #2 Domain: newgateway
  Target SMTP host: our.new.mailgateway.xyz
  settings as above
  
• Rules:
  #1 Name: new-gatway
  Criteria: From contains newdomain.xyz
  Action: Send using route [newgateway]
  Action: Stop rule processing
  
  #2 Name: default
  Criteria: Message size > 0
  Action: Send using route [default]
  keep this rule at the end of the list, it affects every mail that has not been routed by a previous rule.
  
• IP Ranges:
  Delete existing, create one new:
  Name: local
  priority: 5
  Lower and Upper IP: the local IP address of the server
  (our machine is behind NAT, this is important since we basically turn this hMailserver into an open relay)
  Allow connections: SMTP (nothing else)
  Allow deliveries from: ALL
  Require SMTP authentication from: NONE
  
• TCP/IP ports:
  delete all except SMTP, change to custom port (e.g. 1125, 26 or such)

then change your probably only exchange send connector. I could not change the port via ECP so I used PowerShell:

This setup works for us and should work for small sized installations.