No not really in the case of a lost passport which could be seen as been a self inflicted problem.

MPs do have access to a special line to HMPO and would help if there was an actual emergency involved but a work trip isn't an emergency in most cases. But unfortunately the line was a abused a couple of years a go when there was a large backlog and HMPO tightened up the rules on what they would do when contacted by MPs

And a lost passport is not the same as a renewal and has some different procedures and take longer.