You can use a strategy similar to the one I used for my Akumu run. It should get you through unscathed. What you need:
1 Pistol round
3-4 Flash bolts (don't forget there is already one in the room)
Flash bolt upgrade to at least level 3 (I had level 4)
Sprint upgrade to at least level 3 (I had level 4)
At the start, sprint over to the table in the back left corner of the room to dodge an axe. As soon as the zombs start to sprint at you, put a pistol round into the red barrel and light them both on fire. Careful here not to be too close yourself. While they burn, grab the Flash bolt off of the table, and load up the cross bow with it - Stay in your position.
Next, 3 guys will charge in on you. Do not kill any! Instead, wait until they are all through the window and fire off a Flash bolt, stunning all 3. Run to Joseph and stand by him, looking back to the center of the room. Fire off another Flash bolt on the same 3 as they wake back up. You'll notice that no new enemies are entering the room. Keep it that way but not killing any haunted and Joseph will open the door in short order. Note: I had level 4 Flash bolts which I highly recommend for this part, as well as level 4 sprint.
This room is a little more wild. Basically grab as many supplies as you can. When the first few enemies drop down, kite them around the room until a total of 5 are in there. Fire off a Flash bolt and enjoy a moment of down time. As they wake up, fire one off again. Rinse and repeat and Joseph will eventually open the door.
Here, you can get fancy and try to blow up barrels and kill them off. But because of the seemingly random spawning/dropping through the ceiling, this room is very unpredictable and I found it much easier to simply Flash bolt them, even if it took a couple extra traps parts.
One final tip - Once Joseph opens that second door, make sure you have sprint in your sprint meter. Zombs can still hit you as you run down the hall to the exit, and trust me, it's a miserable way to go after all the hassle before it...